{"id":11225,"date":"2026-09-20T05:35:17","date_gmt":"2026-09-20T05:35:17","guid":{"rendered":"https:\/\/otus.ru\/journal\/?p=11225"},"modified":"2026-09-25T15:41:51","modified_gmt":"2026-09-25T15:41:51","slug":"logarifmy-i-ih-osnovnye-svojstva-chto-nuzhno-znat-specialistu","status":"publish","type":"post","link":"https:\/\/otus.ru\/journal\/logarifmy-i-ih-osnovnye-svojstva-chto-nuzhno-znat-specialistu\/","title":{"rendered":"\u041b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u044b \u0438 \u0438\u0445 \u043e\u0441\u043d\u043e\u0432\u043d\u044b\u0435 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430: \u0447\u0442\u043e \u043d\u0443\u0436\u043d\u043e \u0437\u043d\u0430\u0442\u044c \u0441\u043f\u0435\u0446\u0438\u0430\u043b\u0438\u0441\u0442\u0443"},"content":{"rendered":"<p><strong>\u041b\u043e\u0433\u0430\u0440\u0438\u0444\u043c<\/strong> \u0447\u0438\u0441\u043b\u0430 b \u043f\u043e \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u044e a &#8212; \u044d\u0442\u043e \u043f\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c \u0441\u0442\u0435\u043f\u0435\u043d\u0438, \u0432 \u043a\u043e\u0442\u043e\u0440\u0443\u044e \u043d\u0443\u0436\u043d\u043e \u0432\u043e\u0437\u0432\u0435\u0441\u0442\u0438 a, \u0447\u0442\u043e\u0431\u044b \u043f\u043e\u043b\u0443\u0447\u0438\u0442\u044c b; \u0437\u0430\u043f\u0438\u0441\u044b\u0432\u0430\u0435\u0442\u0441\u044f \u043a\u0430\u043a log_a(b). \u0418\u043d\u0430\u0447\u0435 \u0433\u043e\u0432\u043e\u0440\u044f, \u0435\u0441\u043b\u0438 a^x = b, \u0442\u043e x = log_a(b). \u041b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u0435 &#8212; \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u044f, \u043e\u0431\u0440\u0430\u0442\u043d\u0430\u044f \u0432\u043e\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044e \u0432 \u0441\u0442\u0435\u043f\u0435\u043d\u044c: \u043e\u043d\u0430 \u043e\u0442\u0432\u0435\u0447\u0430\u0435\u0442 \u043d\u0430 \u0432\u043e\u043f\u0440\u043e\u0441 \u00ab\u0432 \u043a\u0430\u043a\u0443\u044e \u0441\u0442\u0435\u043f\u0435\u043d\u044c \u0432\u043e\u0437\u0432\u0435\u0441\u0442\u0438 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435, \u0447\u0442\u043e\u0431\u044b \u043f\u043e\u043b\u0443\u0447\u0438\u0442\u044c \u0447\u0438\u0441\u043b\u043e\u00bb.<\/p>\n<div class=\"oj-toc\" style=\"background:#f6f6f6;border-radius:8px;padding:14px 20px;margin:20px 0\">\n<p><strong>\u0421\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435<\/strong><\/p>\n<ol>\n<li><a href=\"#s1\">\u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0438 \u0443\u0441\u043b\u043e\u0432\u0438\u044f<\/a><\/li>\n<li><a href=\"#s2\">\u0422\u0440\u0438 \u043e\u0441\u043d\u043e\u0432\u043d\u044b\u0445 \u0432\u0438\u0434\u0430<\/a><\/li>\n<li><a href=\"#s3\">\u0421\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u043e\u0432: \u0442\u0430\u0431\u043b\u0438\u0446\u0430 \u0441 \u0444\u043e\u0440\u043c\u0443\u043b\u0430\u043c\u0438<\/a><\/li>\n<li><a href=\"#s4\">\u041b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u044b \u0432 Python<\/a><\/li>\n<li><a href=\"#s5\">\u0413\u0434\u0435 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u044b \u0432\u0441\u0442\u0440\u0435\u0447\u0430\u044e\u0442\u0441\u044f \u0432 IT<\/a><\/li>\n<li><a href=\"#s6\">\u0412\u044b\u0432\u043e\u0434\u044b<\/a><\/li>\n<li><a href=\"#s7\">\u0413\u0434\u0435 \u043f\u0440\u0438\u043c\u0435\u043d\u044f\u0435\u0442\u0441\u044f \/ \u0441\u0432\u044f\u0437\u044c \u0441 \u043f\u0440\u0430\u043a\u0442\u0438\u043a\u043e\u0439<\/a><\/li>\n<li><a href=\"#s8\">FAQ<\/a><\/li>\n<\/ol>\n<\/div>\n<p>\u041d\u0438\u0436\u0435 &#8212; \u0442\u043e\u0447\u043d\u043e\u0435 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0441 \u0443\u0441\u043b\u043e\u0432\u0438\u044f\u043c\u0438, \u0442\u0440\u0438 \u043e\u0441\u043d\u043e\u0432\u043d\u044b\u0445 \u0432\u0438\u0434\u0430 (\u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0439, \u0434\u0435\u0441\u044f\u0442\u0438\u0447\u043d\u044b\u0439, \u0434\u0432\u043e\u0438\u0447\u043d\u044b\u0439), \u0442\u0430\u0431\u043b\u0438\u0446\u0430 \u0441\u0432\u043e\u0439\u0441\u0442\u0432 \u0441 \u0444\u043e\u0440\u043c\u0443\u043b\u0430\u043c\u0438 \u0438 \u0440\u0430\u0437\u0431\u043e\u0440 \u0442\u043e\u0433\u043e, \u0437\u0430\u0447\u0435\u043c \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u044b \u043d\u0443\u0436\u043d\u044b \u0432 IT: \u043e\u0446\u0435\u043d\u043a\u0430 \u0441\u043b\u043e\u0436\u043d\u043e\u0441\u0442\u0438 O(log n), \u044d\u043d\u0442\u0440\u043e\u043f\u0438\u044f \u0438 \u0434\u0435\u0446\u0438\u0431\u0435\u043b\u044b. \u041f\u0440\u0438\u043c\u0435\u0440\u044b \u043d\u0430 Python \u043f\u0440\u043e\u0433\u043d\u0430\u043d\u044b, \u0432\u044b\u0432\u043e\u0434 \u043f\u0440\u0438\u0432\u0435\u0434\u0435\u043d \u0440\u044f\u0434\u043e\u043c \u0441 \u043a\u043e\u0434\u043e\u043c.<\/p>\n<h2 id=\"s1\">\u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0438 \u0443\u0441\u043b\u043e\u0432\u0438\u044f<\/h2>\n<p>\u041f\u043e \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044e \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u0435\u0442 \u043d\u0435 \u0434\u043b\u044f \u043b\u044e\u0431\u044b\u0445 \u0447\u0438\u0441\u0435\u043b. \u0422\u0440\u0435\u0431\u043e\u0432\u0430\u043d\u0438\u044f \u0442\u0430\u043a\u0438\u0435:<\/p>\n<ul>\n<li>\u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435 a &gt; 0 \u0438 a \u043d\u0435 \u0440\u0430\u0432\u043d\u043e 1 (\u0435\u0434\u0438\u043d\u0438\u0446\u0430 \u0432 \u043b\u044e\u0431\u043e\u0439 \u0441\u0442\u0435\u043f\u0435\u043d\u0438 \u0434\u0430\u0435\u0442 \u0435\u0434\u0438\u043d\u0438\u0446\u0443, \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u0435 1^x = b \u043d\u0435 \u0438\u043c\u0435\u0435\u0442 \u0440\u0435\u0448\u0435\u043d\u0438\u044f \u043f\u0440\u0438 b, \u043e\u0442\u043b\u0438\u0447\u043d\u043e\u043c \u043e\u0442 1);<\/li>\n<li>\u0430\u0440\u0433\u0443\u043c\u0435\u043d\u0442 b &gt; 0 (\u0432 \u0432\u0435\u0449\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u0445 \u0447\u0438\u0441\u043b\u0430\u0445 \u0441\u0442\u0435\u043f\u0435\u043d\u044c \u043f\u043e\u043b\u043e\u0436\u0438\u0442\u0435\u043b\u044c\u043d\u043e\u0433\u043e \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u044f \u043f\u043e\u043b\u043e\u0436\u0438\u0442\u0435\u043b\u044c\u043d\u0430).<\/li>\n<\/ul>\n<p>\u041f\u0440\u043e\u0432\u0435\u0440\u043a\u0430 \u00ab\u043d\u0430 \u043c\u0435\u0441\u0442\u0435\u00bb: log_2(8) = 3, \u043f\u043e\u0442\u043e\u043c\u0443 \u0447\u0442\u043e 2^3 = 8. \u0410 log_2(0) \u0438\u043b\u0438 log_2(-4) \u0432 \u0432\u0435\u0449\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u0445 \u0447\u0438\u0441\u043b\u0430\u0445 \u043d\u0435 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u044b. \u041a\u043e\u043c\u043f\u043b\u0435\u043a\u0441\u043d\u044b\u0439 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \u0440\u0430\u0441\u0448\u0438\u0440\u044f\u0435\u0442 \u043e\u0431\u043b\u0430\u0441\u0442\u044c, \u043d\u043e \u0434\u043b\u044f \u043f\u0440\u0438\u043a\u043b\u0430\u0434\u043d\u044b\u0445 \u0437\u0430\u0434\u0430\u0447 \u0432 IT \u0440\u0430\u0431\u043e\u0442\u0430\u044e\u0442 \u0438\u043c\u0435\u043d\u043d\u043e \u0441 \u0432\u0435\u0449\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u043c \u0441\u043b\u0443\u0447\u0430\u0435\u043c, \u0438 \u044d\u0442\u043e \u0443\u043f\u0440\u043e\u0449\u0435\u043d\u0438\u0435 \u0434\u043e\u0441\u0442\u0430\u0442\u043e\u0447\u043d\u043e \u043f\u043e\u0447\u0442\u0438 \u0432\u0441\u0435\u0433\u0434\u0430.<\/p>\n<p>\u041b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \u043f\u043e\u043b\u0435\u0437\u0435\u043d \u0442\u0430\u043c, \u0433\u0434\u0435 \u0432\u0435\u043b\u0438\u0447\u0438\u043d\u044b \u043e\u0442\u043b\u0438\u0447\u0430\u044e\u0442\u0441\u044f \u0432 \u0440\u0430\u0437\u044b \u0438 \u043d\u0430 \u043f\u043e\u0440\u044f\u0434\u043a\u0438: \u043e\u043d \u0441\u0436\u0438\u043c\u0430\u0435\u0442 \u0442\u0430\u043a\u043e\u0439 \u0440\u0430\u0437\u0431\u0440\u043e\u0441 \u0432 \u0443\u0434\u043e\u0431\u043d\u0443\u044e \u0448\u043a\u0430\u043b\u0443. \u0418\u043c\u0435\u043d\u043d\u043e \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0438\u0447\u0435\u0441\u043a\u0438\u0435 \u0448\u043a\u0430\u043b\u044b \u043f\u0440\u0438\u043c\u0435\u043d\u044f\u044e\u0442 \u0434\u043b\u044f \u0433\u0440\u043e\u043c\u043a\u043e\u0441\u0442\u0438, \u0447\u0430\u0441\u0442\u043e\u0442 \u0438 \u043e\u0431\u044a\u0435\u043c\u043e\u0432 \u0434\u0430\u043d\u043d\u044b\u0445.<\/p>\n<h2 id=\"s2\">\u0422\u0440\u0438 \u043e\u0441\u043d\u043e\u0432\u043d\u044b\u0445 \u0432\u0438\u0434\u0430<\/h2>\n<p>\u041b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u044b \u0440\u0430\u0437\u043b\u0438\u0447\u0430\u044e\u0442 \u043f\u043e \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u044e. \u0422\u0440\u0438 \u0432\u0438\u0434\u0430 \u043f\u043e\u043a\u0440\u044b\u0432\u0430\u044e\u0442 \u043f\u043e\u0447\u0442\u0438 \u0432\u0441\u0435 \u043f\u0440\u0438\u043a\u043b\u0430\u0434\u043d\u044b\u0435 \u0437\u0430\u0434\u0430\u0447\u0438.<\/p>\n<table>\n<thead>\n<tr>\n<th>\u0412\u0438\u0434<\/th>\n<th>\u041e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435<\/th>\n<th>\u0417\u0430\u043f\u0438\u0441\u044c<\/th>\n<th>\u0413\u0434\u0435 \u043d\u0443\u0436\u0435\u043d<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>\u041d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0439<\/td>\n<td>e (\u043f\u0440\u0438\u043c\u0435\u0440\u043d\u043e 2.71828)<\/td>\n<td>ln(x) \u0438\u043b\u0438 log_e(x)<\/td>\n<td>\u043c\u0430\u0442\u0430\u043d\u0430\u043b\u0438\u0437, \u043f\u0440\u043e\u0446\u0435\u0441\u0441\u044b \u0440\u043e\u0441\u0442\u0430 \u0438 \u0437\u0430\u0442\u0443\u0445\u0430\u043d\u0438\u044f, ML<\/td>\n<\/tr>\n<tr>\n<td>\u0414\u0435\u0441\u044f\u0442\u0438\u0447\u043d\u044b\u0439<\/td>\n<td>10<\/td>\n<td>lg(x) \u0438\u043b\u0438 log10(x)<\/td>\n<td>\u043f\u043e\u0440\u044f\u0434\u043a\u0438 \u0432\u0435\u043b\u0438\u0447\u0438\u043d, \u0434\u0435\u0446\u0438\u0431\u0435\u043b\u044b, \u0448\u043a\u0430\u043b\u0430 pH<\/td>\n<\/tr>\n<tr>\n<td>\u0414\u0432\u043e\u0438\u0447\u043d\u044b\u0439<\/td>\n<td>2<\/td>\n<td>log2(x)<\/td>\n<td>\u0441\u043b\u043e\u0436\u043d\u043e\u0441\u0442\u044c \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c\u043e\u0432, \u044d\u043d\u0442\u0440\u043e\u043f\u0438\u044f, \u0431\u0438\u0442\u044b<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>\u041d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0439 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c<\/strong> \u0438\u043c\u0435\u0435\u0442 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435\u043c \u0447\u0438\u0441\u043b\u043e \u042d\u0439\u043b\u0435\u0440\u0430 e &#8212; \u0438\u0440\u0440\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u044c\u043d\u0443\u044e \u043a\u043e\u043d\u0441\u0442\u0430\u043d\u0442\u0443, \u043f\u0440\u0438\u0431\u043b\u0438\u0437\u0438\u0442\u0435\u043b\u044c\u043d\u043e 2.71828. \u041e\u043d\u043e \u0432\u043e\u0437\u043d\u0438\u043a\u0430\u0435\u0442 \u043a\u0430\u043a \u043f\u0440\u0435\u0434\u0435\u043b \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u043e\u0433\u043e \u0440\u043e\u0441\u0442\u0430: \u0435\u0441\u043b\u0438 \u0432\u0435\u043b\u0438\u0447\u0438\u043d\u0430 \u0440\u0430\u0441\u0442\u0435\u0442 \u043f\u0440\u043e\u043f\u043e\u0440\u0446\u0438\u043e\u043d\u0430\u043b\u044c\u043d\u043e \u0441\u0432\u043e\u0435\u043c\u0443 \u0442\u0435\u043a\u0443\u0449\u0435\u043c\u0443 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044e, \u0437\u0430 \u0435\u0434\u0438\u043d\u0438\u0446\u0443 \u0432\u0440\u0435\u043c\u0435\u043d\u0438 \u043e\u043d\u0430 \u0443\u043c\u043d\u043e\u0436\u0430\u0435\u0442\u0441\u044f \u043d\u0430 e. \u041f\u043e \u0442\u0430\u043a\u043e\u043c\u0443 \u0437\u0430\u043a\u043e\u043d\u0443 \u043c\u043e\u0434\u0435\u043b\u0438\u0440\u0443\u044e\u0442 \u0440\u0430\u0437\u043c\u043d\u043e\u0436\u0435\u043d\u0438\u0435 \u0431\u0430\u043a\u0442\u0435\u0440\u0438\u0439, \u0440\u0430\u0437\u0440\u044f\u0434 \u043a\u043e\u043d\u0434\u0435\u043d\u0441\u0430\u0442\u043e\u0440\u0430, \u043e\u0441\u0442\u044b\u0432\u0430\u043d\u0438\u0435 \u0442\u0435\u043b\u0430.<\/p>\n<p><strong>\u0414\u0435\u0441\u044f\u0442\u0438\u0447\u043d\u044b\u0439 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c<\/strong> \u0443\u0434\u043e\u0431\u0435\u043d \u0434\u043b\u044f \u043e\u0446\u0435\u043d\u043a\u0438 \u043f\u043e\u0440\u044f\u0434\u043a\u0430 \u0432\u0435\u043b\u0438\u0447\u0438\u043d\u044b: lg(1000) = 3 \u043e\u0437\u043d\u0430\u0447\u0430\u0435\u0442 \u00ab\u0442\u0440\u0438 \u043d\u0443\u043b\u044f\u00bb. <strong>\u0414\u0432\u043e\u0438\u0447\u043d\u044b\u0439<\/strong> \u043e\u0442\u0432\u0435\u0447\u0430\u0435\u0442 \u043d\u0430 \u0432\u043e\u043f\u0440\u043e\u0441 \u00ab\u0441\u043a\u043e\u043b\u044c\u043a\u043e \u0440\u0430\u0437 \u0447\u0438\u0441\u043b\u043e \u0434\u0435\u043b\u0438\u0442\u0441\u044f \u043f\u043e\u043f\u043e\u043b\u0430\u043c\u00bb \u0438 \u043f\u043e\u0442\u043e\u043c\u0443 \u0435\u0441\u0442\u0435\u0441\u0442\u0432\u0435\u043d\u0435\u043d \u0432 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0435, \u0433\u0434\u0435 \u0434\u0430\u043d\u043d\u044b\u0435 \u0438\u0437\u043c\u0435\u0440\u044f\u044e\u0442 \u0432 \u0431\u0438\u0442\u0430\u0445, \u0430 \u043a\u043e\u043c\u043f\u044c\u044e\u0442\u0435\u0440\u044b \u0440\u0430\u0431\u043e\u0442\u0430\u044e\u0442 \u0432 \u0434\u0432\u043e\u0438\u0447\u043d\u043e\u0439 \u0441\u0438\u0441\u0442\u0435\u043c\u0435.<\/p>\n<p>\u041d\u0435 \u043f\u0443\u0442\u0430\u0439\u0442\u0435 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435 \u0438 \u0430\u0440\u0433\u0443\u043c\u0435\u043d\u0442: \u0432 log_a(b) \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435 a \u0441\u0442\u043e\u0438\u0442 \u0432\u043d\u0438\u0437\u0443 \u0438 \u0437\u0430\u0434\u0430\u0435\u0442 \u00ab\u0448\u0430\u0433\u00bb \u0448\u043a\u0430\u043b\u044b, \u0430 b &#8212; \u0447\u0438\u0441\u043b\u043e, \u0434\u043b\u044f \u043a\u043e\u0442\u043e\u0440\u043e\u0433\u043e \u0438\u0449\u0443\u0442 \u043f\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c. \u0421\u043c\u0435\u043d\u0430 \u0440\u043e\u043b\u0435\u0439 \u043c\u0435\u043d\u044f\u0435\u0442 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442: log_2(8) = 3, \u043d\u043e log_8(2) \u043f\u0440\u0438\u043c\u0435\u0440\u043d\u043e 0.333.<\/p>\n<h2 id=\"s3\">\u0421\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u043e\u0432: \u0442\u0430\u0431\u043b\u0438\u0446\u0430 \u0441 \u0444\u043e\u0440\u043c\u0443\u043b\u0430\u043c\u0438<\/h2>\n<p>\u0421\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u043d\u0438\u0436\u0435 \u0432\u044b\u0432\u043e\u0434\u044f\u0442\u0441\u044f \u0438\u0437 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u0438 \u0440\u0430\u0431\u043e\u0442\u0430\u044e\u0442 \u0432 \u043e\u0431\u0435 \u0441\u0442\u043e\u0440\u043e\u043d\u044b &#8212; \u0438 \u0434\u043b\u044f \u0443\u043f\u0440\u043e\u0449\u0435\u043d\u0438\u044f \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u044f, \u0438 \u0434\u043b\u044f \u0435\u0433\u043e \u0441\u0432\u0435\u0440\u0442\u043a\u0438. \u0412\u0441\u0435 \u043e\u043d\u0438 \u0442\u0440\u0435\u0431\u0443\u044e\u0442, \u0447\u0442\u043e\u0431\u044b \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u044f \u0438 \u0430\u0440\u0433\u0443\u043c\u0435\u043d\u0442\u044b \u0443\u0434\u043e\u0432\u043b\u0435\u0442\u0432\u043e\u0440\u044f\u043b\u0438 \u0443\u0441\u043b\u043e\u0432\u0438\u044f\u043c \u043e\u0431\u043b\u0430\u0441\u0442\u0438 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f.<\/p>\n<table>\n<thead>\n<tr>\n<th>\u0421\u0432\u043e\u0439\u0441\u0442\u0432\u043e<\/th>\n<th>\u0424\u043e\u0440\u043c\u0443\u043b\u0430<\/th>\n<th>\u0423\u0441\u043b\u043e\u0432\u0438\u0435<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>\u041b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \u0435\u0434\u0438\u043d\u0438\u0446\u044b<\/td>\n<td>log_a(1) = 0<\/td>\n<td>a &gt; 0, a \u043d\u0435 \u0440\u0430\u0432\u043d\u043e 1<\/td>\n<\/tr>\n<tr>\n<td>\u041b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u044f<\/td>\n<td>log_a(a) = 1<\/td>\n<td>a &gt; 0, a \u043d\u0435 \u0440\u0430\u0432\u043d\u043e 1<\/td>\n<\/tr>\n<tr>\n<td>\u041b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f<\/td>\n<td>log_a(x*y) = log_a(x) + log_a(y)<\/td>\n<td>x &gt; 0, y &gt; 0<\/td>\n<\/tr>\n<tr>\n<td>\u041b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \u0447\u0430\u0441\u0442\u043d\u043e\u0433\u043e<\/td>\n<td>log_a(x\/y) = log_a(x) &#8212; log_a(y)<\/td>\n<td>x &gt; 0, y &gt; 0<\/td>\n<\/tr>\n<tr>\n<td>\u041b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \u0441\u0442\u0435\u043f\u0435\u043d\u0438<\/td>\n<td>log_a(x^p) = p * log_a(x)<\/td>\n<td>x &gt; 0<\/td>\n<\/tr>\n<tr>\n<td>\u0421\u0442\u0435\u043f\u0435\u043d\u044c \u0432 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0438<\/td>\n<td>log_(a^k)(x) = (1\/k) * log_a(x)<\/td>\n<td>k \u043d\u0435 \u0440\u0430\u0432\u043d\u043e 0<\/td>\n<\/tr>\n<tr>\n<td>\u041f\u0435\u0440\u0435\u0445\u043e\u0434 \u043a \u043d\u043e\u0432\u043e\u043c\u0443 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u044e<\/td>\n<td>log_a(b) = log_c(b) \/ log_c(a)<\/td>\n<td>c &gt; 0, c \u043d\u0435 \u0440\u0430\u0432\u043d\u043e 1<\/td>\n<\/tr>\n<tr>\n<td>\u041e\u0431\u043c\u0435\u043d \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u044f \u0438 \u0430\u0440\u0433\u0443\u043c\u0435\u043d\u0442\u0430<\/td>\n<td>log_a(b) = 1 \/ log_b(a)<\/td>\n<td>b &gt; 0, b \u043d\u0435 \u0440\u0430\u0432\u043d\u043e 1<\/td>\n<\/tr>\n<tr>\n<td>\u041e\u0441\u043d\u043e\u0432\u043d\u043e\u0435 \u0442\u043e\u0436\u0434\u0435\u0441\u0442\u0432\u043e<\/td>\n<td>a^(log_a(b)) = b<\/td>\n<td>b &gt; 0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u041a\u043b\u044e\u0447\u0435\u0432\u043e\u0435 \u0438\u0437 \u043f\u0440\u0430\u043a\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0445 &#8212; <strong>\u043f\u0435\u0440\u0435\u0445\u043e\u0434 \u043a \u043d\u043e\u0432\u043e\u043c\u0443 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u044e<\/strong>. \u0421\u0442\u0430\u043d\u0434\u0430\u0440\u0442\u043d\u044b\u0435 \u0431\u0438\u0431\u043b\u0438\u043e\u0442\u0435\u043a\u0438 \u0440\u0435\u0434\u043a\u043e \u0434\u0430\u044e\u0442 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \u043f\u043e \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u043b\u044c\u043d\u043e\u043c\u0443 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u044e \u043d\u0430\u043f\u0440\u044f\u043c\u0443\u044e, \u0437\u0430\u0442\u043e \u043f\u043e\u0447\u0442\u0438 \u0432\u0441\u0435\u0433\u0434\u0430 \u0435\u0441\u0442\u044c ln. \u041f\u043e\u044d\u0442\u043e\u043c\u0443 log_a(b) \u0441\u0447\u0438\u0442\u0430\u044e\u0442 \u043a\u0430\u043a ln(b) \/ ln(a). \u041d\u0430\u043f\u0440\u0438\u043c\u0435\u0440, log_2(x) = ln(x) \/ ln(2).<\/p>\n<p>\u0424\u043e\u0440\u043c\u0443\u043b\u0430 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f \u043e\u0431\u044a\u044f\u0441\u043d\u044f\u0435\u0442 \u0441\u0442\u0430\u0440\u044b\u0439 \u0438\u043d\u0436\u0435\u043d\u0435\u0440\u043d\u044b\u0439 \u043f\u0440\u0438\u0435\u043c: \u0443\u043c\u043d\u043e\u0436\u0435\u043d\u0438\u0435 \u0447\u0438\u0441\u0435\u043b \u0441\u0432\u043e\u0434\u0438\u043b\u0438 \u043a \u0441\u043b\u043e\u0436\u0435\u043d\u0438\u044e \u0438\u0445 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u043e\u0432 \u043f\u043e \u0442\u0430\u0431\u043b\u0438\u0446\u0430\u043c. \u0422\u043e\u0442 \u0436\u0435 \u043f\u0440\u0438\u043d\u0446\u0438\u043f \u0440\u0430\u0431\u043e\u0442\u0430\u0435\u0442 \u0432 \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f\u0445 \u0441 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u044f\u043c\u0438 &#8212; \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u0430 \u043c\u0430\u043b\u044b\u0445 \u0432\u0435\u043b\u0438\u0447\u0438\u043d \u0437\u0430\u043c\u0435\u043d\u044f\u044e\u0442 \u0441\u0443\u043c\u043c\u043e\u0439 \u0438\u0445 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u043e\u0432, \u0447\u0442\u043e\u0431\u044b \u043d\u0435 \u0442\u0435\u0440\u044f\u0442\u044c \u0442\u043e\u0447\u043d\u043e\u0441\u0442\u044c (log-likelihood \u0432 \u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u043a\u0435 \u0438 ML).<\/p>\n<h2 id=\"s4\">\u041b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u044b \u0432 Python<\/h2>\n<p>\u0412 Python \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u044b \u0436\u0438\u0432\u0443\u0442 \u0432 \u043c\u043e\u0434\u0443\u043b\u0435 math \u0441\u0442\u0430\u043d\u0434\u0430\u0440\u0442\u043d\u043e\u0439 \u0431\u0438\u0431\u043b\u0438\u043e\u0442\u0435\u043a\u0438, \u043e\u0442\u0434\u0435\u043b\u044c\u043d\u043e\u0439 \u0443\u0441\u0442\u0430\u043d\u043e\u0432\u043a\u0438 \u043d\u0435 \u043d\u0443\u0436\u043d\u043e. \u041d\u0438\u0436\u0435 &#8212; \u043c\u0438\u043d\u0438\u043c\u0430\u043b\u044c\u043d\u044b\u0439 \u0440\u0430\u0431\u043e\u0447\u0438\u0439 \u043f\u0440\u0438\u043c\u0435\u0440: \u0441\u043a\u043e\u043f\u0438\u0440\u0443\u0439\u0442\u0435 \u0438 \u0437\u0430\u043f\u0443\u0441\u0442\u0438\u0442\u0435 \u0446\u0435\u043b\u0438\u043a\u043e\u043c, \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442 \u0432 \u043a\u043e\u043c\u043c\u0435\u043d\u0442\u0430\u0440\u0438\u044f\u0445.<\/p>\n<pre style=\"background:#282c34;color:#abb2bf;border-radius:8px;padding:16px 20px;margin:20px 0;overflow-x:auto;font-size:14px;line-height:1.55;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\"><code style=\"background:none\"><span style=\"color: #C678DD\">import<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #E06C75\">math<\/span>\n\n<span style=\"color: #E5C07B\">print<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">math<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">log<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #D19A66\">8<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">2<\/span><span style=\"color: #ABB2BF\">))<\/span>     <span style=\"color: #7F848E\"># 3.0   - log \u043f\u043e \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u044e: \u0430\u0440\u0433\u0443\u043c\u0435\u043d\u0442, \u043f\u043e\u0442\u043e\u043c \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435<\/span>\n<span style=\"color: #E5C07B\">print<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">math<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">log2<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #D19A66\">8<\/span><span style=\"color: #ABB2BF\">))<\/span>       <span style=\"color: #7F848E\"># 3.0   - \u0441\u043f\u0435\u0446\u0438\u0430\u043b\u0438\u0437\u0438\u0440\u043e\u0432\u0430\u043d\u043d\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0434\u043b\u044f \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u044f 2<\/span>\n<span style=\"color: #E5C07B\">print<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">math<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">log10<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #D19A66\">1000<\/span><span style=\"color: #ABB2BF\">))<\/span>   <span style=\"color: #7F848E\"># 3.0   - \u0434\u0435\u0441\u044f\u0442\u0438\u0447\u043d\u044b\u0439 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c<\/span>\n<span style=\"color: #E5C07B\">print<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">math<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">log<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">math<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">e<\/span><span style=\"color: #ABB2BF\">))<\/span>   <span style=\"color: #7F848E\"># 1.0   - \u0431\u0435\u0437 \u0432\u0442\u043e\u0440\u043e\u0433\u043e \u0430\u0440\u0433\u0443\u043c\u0435\u043d\u0442\u0430 = \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0439 ln<\/span><\/code><\/pre>\n<p>\u041e\u0431\u0440\u0430\u0442\u0438\u0442\u0435 \u0432\u043d\u0438\u043c\u0430\u043d\u0438\u0435 \u043d\u0430 \u043f\u043e\u0440\u044f\u0434\u043e\u043a \u0430\u0440\u0433\u0443\u043c\u0435\u043d\u0442\u043e\u0432: <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">math.log(x, base)<\/code> \u043f\u0440\u0438\u043d\u0438\u043c\u0430\u0435\u0442 \u0441\u043d\u0430\u0447\u0430\u043b\u0430 \u0447\u0438\u0441\u043b\u043e, \u043f\u043e\u0442\u043e\u043c \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435 &#8212; \u044d\u0442\u043e \u043f\u0440\u043e\u0442\u0438\u0432\u043e\u043f\u043e\u043b\u043e\u0436\u043d\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u0437\u0430\u043f\u0438\u0441\u0438 log_base(x). \u0424\u0443\u043d\u043a\u0446\u0438\u044f \u0432\u0441\u0435\u0433\u0434\u0430 \u0432\u043e\u0437\u0432\u0440\u0430\u0449\u0430\u0435\u0442 float, \u0434\u0430\u0436\u0435 \u043a\u043e\u0433\u0434\u0430 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442 \u0446\u0435\u043b\u044b\u0439.<\/p>\n<p>\u0414\u043b\u044f \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0439 2 \u0438 10 \u0435\u0441\u0442\u044c \u043e\u0442\u0434\u0435\u043b\u044c\u043d\u044b\u0435 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">log2<\/code> \u0438 <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">log10<\/code>, \u0438 \u044d\u0442\u043e \u043d\u0435 \u043f\u0440\u043e\u0441\u0442\u043e \u0441\u0430\u0445\u0430\u0440. \u041e\u043d\u0438 \u0442\u043e\u0447\u043d\u0435\u0435 \u0443\u043d\u0438\u0432\u0435\u0440\u0441\u0430\u043b\u044c\u043d\u043e\u0433\u043e <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">math.log(x, base)<\/code>, \u0433\u0434\u0435 \u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0434\u0432\u0443\u0445 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u043e\u0432 \u043d\u0430\u043a\u0430\u043f\u043b\u0438\u0432\u0430\u0435\u0442 \u043e\u0448\u0438\u0431\u043a\u0443 float:<\/p>\n<pre style=\"background:#282c34;color:#abb2bf;border-radius:8px;padding:16px 20px;margin:20px 0;overflow-x:auto;font-size:14px;line-height:1.55;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\"><code style=\"background:none\"><span style=\"color: #C678DD\">import<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #E06C75\">math<\/span>\n\n<span style=\"color: #E5C07B\">print<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">math<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">log<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #D19A66\">1000<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">10<\/span><span style=\"color: #ABB2BF\">))<\/span>  <span style=\"color: #7F848E\"># 2.9999999999999996 - \u0434\u0435\u043b\u0435\u043d\u0438\u0435 ln(1000)\/ln(10), \u043e\u0448\u0438\u0431\u043a\u0430 float<\/span>\n<span style=\"color: #E5C07B\">print<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">math<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">log10<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #D19A66\">1000<\/span><span style=\"color: #ABB2BF\">))<\/span>    <span style=\"color: #7F848E\"># 3.0 - \u0441\u043f\u0435\u0446\u0438\u0430\u043b\u0438\u0437\u0438\u0440\u043e\u0432\u0430\u043d\u043d\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0434\u0430\u0435\u0442 \u0442\u043e\u0447\u043d\u044b\u0439 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442<\/span><\/code><\/pre>\n<p>\u0412\u044b\u0432\u043e\u0434 <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">2.9999999999999996<\/code> &#8212; \u043d\u0435 \u0431\u0430\u0433, \u0430 \u0441\u043b\u0435\u0434\u0441\u0442\u0432\u0438\u0435 \u0434\u0432\u043e\u0438\u0447\u043d\u043e\u0433\u043e \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u0435\u043d\u0438\u044f \u0432\u0435\u0449\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u0445 \u0447\u0438\u0441\u0435\u043b. \u041f\u0440\u0430\u0432\u0438\u043b\u043e \u043f\u0440\u043e\u0441\u0442\u043e\u0435: \u0435\u0441\u043b\u0438 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435 2 \u0438\u043b\u0438 10, \u0431\u0435\u0440\u0438\u0442\u0435 <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">log2<\/code>\/<code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">log10<\/code>, \u0430 <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">math.log(x, base)<\/code> \u043e\u0441\u0442\u0430\u0432\u044c\u0442\u0435 \u0434\u043b\u044f \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u043b\u044c\u043d\u044b\u0445 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0439.<\/p>\n<p>\u041e\u0442\u0434\u0435\u043b\u044c\u043d\u043e \u0441\u0442\u043e\u0438\u0442 \u0444\u0443\u043d\u043a\u0446\u0438\u044f <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">math.log1p(x)<\/code> &#8212; \u043e\u043d\u0430 \u0441\u0447\u0438\u0442\u0430\u0435\u0442 ln(1 + x) \u0438 \u0442\u043e\u0447\u043d\u0435\u0435 \u043f\u0440\u044f\u043c\u043e\u0433\u043e \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u044f, \u043a\u043e\u0433\u0434\u0430 x \u0431\u043b\u0438\u0437\u043e\u043a \u043a \u043d\u0443\u043b\u044e:<\/p>\n<pre style=\"background:#282c34;color:#abb2bf;border-radius:8px;padding:16px 20px;margin:20px 0;overflow-x:auto;font-size:14px;line-height:1.55;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\"><code style=\"background:none\"><span style=\"color: #C678DD\">import<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #E06C75\">math<\/span>\n\n<span style=\"color: #E06C75\">x<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #D19A66\">1e-15<\/span>\n<span style=\"color: #E5C07B\">print<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">math<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">log<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #D19A66\">1<\/span> <span style=\"color: #56B6C2\">+<\/span> <span style=\"color: #E06C75\">x<\/span><span style=\"color: #ABB2BF\">))<\/span>  <span style=\"color: #7F848E\"># 1.110223024625156e-15 - \u0441\u0443\u043c\u043c\u0430 1+x \u0443\u0436\u0435 \u0442\u0435\u0440\u044f\u0435\u0442 \u0442\u043e\u0447\u043d\u043e\u0441\u0442\u044c<\/span>\n<span style=\"color: #E5C07B\">print<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">math<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">log1p<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">x<\/span><span style=\"color: #ABB2BF\">))<\/span>    <span style=\"color: #7F848E\"># 9.999999999999995e-16 - \u0431\u043b\u0438\u0436\u0435 \u043a \u0438\u0441\u0442\u0438\u043d\u043d\u043e\u043c\u0443 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044e<\/span><\/code><\/pre>\n<p>\u0417\u0434\u0435\u0441\u044c <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">1 + x<\/code> \u043f\u0440\u0438 \u043e\u0447\u0435\u043d\u044c \u043c\u0430\u043b\u043e\u043c x \u043e\u043a\u0440\u0443\u0433\u043b\u044f\u0435\u0442\u0441\u044f \u0435\u0449\u0435 \u0434\u043e \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u044f, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">log1p<\/code> \u0434\u0430\u0435\u0442 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442 \u0431\u043b\u0438\u0436\u0435 \u043a \u043d\u0430\u0441\u0442\u043e\u044f\u0449\u0435\u043c\u0443 ln(1 + x). \u041d\u0430 \u043f\u0440\u0430\u043a\u0442\u0438\u043a\u0435 \u044d\u0442\u043e \u043d\u0443\u0436\u043d\u043e \u0432 \u0444\u0438\u043d\u0430\u043d\u0441\u043e\u0432\u044b\u0445 \u0438 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u043d\u044b\u0445 \u0440\u0430\u0441\u0447\u0435\u0442\u0430\u0445 \u0441 \u043c\u0430\u043b\u044b\u043c\u0438 \u043f\u0440\u0438\u0440\u0430\u0449\u0435\u043d\u0438\u044f\u043c\u0438.<\/p>\n<h2 id=\"s5\">\u0413\u0434\u0435 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u044b \u0432\u0441\u0442\u0440\u0435\u0447\u0430\u044e\u0442\u0441\u044f \u0432 IT<\/h2>\n<p>\u0421\u043b\u043e\u0436\u043d\u043e\u0441\u0442\u044c \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c\u043e\u0432. \u0417\u0430\u043f\u0438\u0441\u044c O(log n) \u043e\u0437\u043d\u0430\u0447\u0430\u0435\u0442, \u0447\u0442\u043e \u0447\u0438\u0441\u043b\u043e \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u0439 \u0440\u0430\u0441\u0442\u0435\u0442 \u043a\u0430\u043a \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \u043e\u0442 \u0440\u0430\u0437\u043c\u0435\u0440\u0430 \u0432\u0445\u043e\u0434\u0430: \u0443\u0434\u0432\u043e\u0435\u043d\u0438\u0435 \u0434\u0430\u043d\u043d\u044b\u0445 \u0434\u043e\u0431\u0430\u0432\u043b\u044f\u0435\u0442 \u043b\u0438\u0448\u044c \u043e\u0434\u0438\u043d \u0448\u0430\u0433. \u041a\u043b\u0430\u0441\u0441\u0438\u043a\u0430 &#8212; \u0431\u0438\u043d\u0430\u0440\u043d\u044b\u0439 \u043f\u043e\u0438\u0441\u043a, \u043a\u043e\u0442\u043e\u0440\u044b\u0439 \u043d\u0430 \u043a\u0430\u0436\u0434\u043e\u043c \u0448\u0430\u0433\u0435 \u0434\u0435\u043b\u0438\u0442 \u0434\u0438\u0430\u043f\u0430\u0437\u043e\u043d \u043f\u043e\u043f\u043e\u043b\u0430\u043c. \u0414\u043b\u044f \u043c\u0438\u043b\u043b\u0438\u043e\u043d\u0430 \u044d\u043b\u0435\u043c\u0435\u043d\u0442\u043e\u0432 \u0445\u0432\u0430\u0442\u0438\u0442 \u043e\u043a\u043e\u043b\u043e \u0434\u0432\u0430\u0434\u0446\u0430\u0442\u0438 \u0441\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u0439:<\/p>\n<pre style=\"background:#282c34;color:#abb2bf;border-radius:8px;padding:16px 20px;margin:20px 0;overflow-x:auto;font-size:14px;line-height:1.55;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\"><code style=\"background:none\"><span style=\"color: #C678DD\">import<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #E06C75\">math<\/span>\n\n<span style=\"color: #E5C07B\">print<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">math<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">log2<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #D19A66\">1_000_000<\/span><span style=\"color: #ABB2BF\">))<\/span>            <span style=\"color: #7F848E\"># 19.931568569324174<\/span>\n<span style=\"color: #E5C07B\">print<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">math<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">ceil<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">math<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">log2<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #D19A66\">1_000_000<\/span><span style=\"color: #ABB2BF\">)))<\/span> <span style=\"color: #7F848E\"># 20 - \u043c\u0430\u043a\u0441\u0438\u043c\u0443\u043c \u0448\u0430\u0433\u043e\u0432 \u0431\u0438\u043d\u0430\u0440\u043d\u043e\u0433\u043e \u043f\u043e\u0438\u0441\u043a\u0430<\/span><\/code><\/pre>\n<p>\u041f\u043e \u044d\u0442\u043e\u043c\u0443 \u0436\u0435 \u0437\u0430\u043a\u043e\u043d\u0443 \u0440\u0430\u0431\u043e\u0442\u0430\u044e\u0442 \u0441\u0431\u0430\u043b\u0430\u043d\u0441\u0438\u0440\u043e\u0432\u0430\u043d\u043d\u044b\u0435 \u0434\u0435\u0440\u0435\u0432\u044c\u044f \u043f\u043e\u0438\u0441\u043a\u0430 \u0438 \u0432\u044b\u0441\u043e\u0442\u0430 \u0434\u0432\u043e\u0438\u0447\u043d\u043e\u0433\u043e \u0434\u0435\u0440\u0435\u0432\u0430: log2(n) &#8212; \u043f\u0440\u0438\u043c\u0435\u0440\u043d\u0430\u044f \u0433\u043b\u0443\u0431\u0438\u043d\u0430. \u041e\u0442\u0441\u044e\u0434\u0430 \u043f\u0440\u0430\u043a\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0439 \u0432\u044b\u0432\u043e\u0434 &#8212; \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c \u0441 O(log n) \u043c\u0430\u0441\u0448\u0442\u0430\u0431\u0438\u0440\u0443\u0435\u0442\u0441\u044f \u043d\u0435\u0441\u043e\u043f\u043e\u0441\u0442\u0430\u0432\u0438\u043c\u043e \u043b\u0443\u0447\u0448\u0435 \u043b\u0438\u043d\u0435\u0439\u043d\u043e\u0433\u043e O(n).<\/p>\n<p>\u042d\u043d\u0442\u0440\u043e\u043f\u0438\u044f \u0438 \u0431\u0438\u0442\u044b. \u041a\u043e\u043b\u0438\u0447\u0435\u0441\u0442\u0432\u043e \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u0438 \u0438\u0437\u043c\u0435\u0440\u044f\u044e\u0442 \u0447\u0435\u0440\u0435\u0437 \u0434\u0432\u043e\u0438\u0447\u043d\u044b\u0439 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c: \u0447\u0442\u043e\u0431\u044b \u0437\u0430\u043a\u043e\u0434\u0438\u0440\u043e\u0432\u0430\u0442\u044c \u043e\u0434\u0438\u043d \u0438\u0437 N \u0440\u0430\u0432\u043d\u043e\u0432\u0435\u0440\u043e\u044f\u0442\u043d\u044b\u0445 \u0438\u0441\u0445\u043e\u0434\u043e\u0432, \u043d\u0443\u0436\u043d\u043e log2(N) \u0431\u0438\u0442. \u0414\u043b\u044f \u0447\u0435\u0441\u0442\u043d\u043e\u0439 \u043c\u043e\u043d\u0435\u0442\u044b \u044d\u0442\u043e log2(2) = 1 \u0431\u0438\u0442, \u0434\u043b\u044f \u0438\u0433\u0440\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u043a\u0443\u0431\u0438\u043a\u0430 &#8212; log2(6), \u043f\u0440\u0438\u043c\u0435\u0440\u043d\u043e 2.585 \u0431\u0438\u0442. \u041d\u0430 \u044d\u0442\u043e\u0439 \u0444\u043e\u0440\u043c\u0443\u043b\u0435 \u0441\u0442\u043e\u044f\u0442 \u043e\u0446\u0435\u043d\u043a\u0438 \u0441\u0436\u0430\u0442\u0438\u044f \u0434\u0430\u043d\u043d\u044b\u0445 \u0438 \u044d\u043d\u0442\u0440\u043e\u043f\u0438\u044f \u0428\u0435\u043d\u043d\u043e\u043d\u0430.<\/p>\n<p>\u0414\u0435\u0446\u0438\u0431\u0435\u043b\u044b \u0438 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0438\u0447\u0435\u0441\u043a\u0438\u0435 \u0448\u043a\u0430\u043b\u044b. \u0413\u0440\u043e\u043c\u043a\u043e\u0441\u0442\u044c, \u043e\u0442\u043d\u043e\u0448\u0435\u043d\u0438\u0435 \u0441\u0438\u0433\u043d\u0430\u043b\/\u0448\u0443\u043c, \u043a\u043e\u044d\u0444\u0444\u0438\u0446\u0438\u0435\u043d\u0442\u044b \u0443\u0441\u0438\u043b\u0435\u043d\u0438\u044f \u0432\u044b\u0440\u0430\u0436\u0430\u044e\u0442 \u0432 \u0434\u0435\u0446\u0438\u0431\u0435\u043b\u0430\u0445 \u0447\u0435\u0440\u0435\u0437 \u0434\u0435\u0441\u044f\u0442\u0438\u0447\u043d\u044b\u0439 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c: \u0432\u0435\u043b\u0438\u0447\u0438\u043d\u0430 \u0432 \u0434\u0411 \u0440\u0430\u0432\u043d\u0430 10 * lg(P1 \/ P0). \u041b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0438\u0447\u0435\u0441\u043a\u0430\u044f \u0448\u043a\u0430\u043b\u0430 \u0441\u0436\u0438\u043c\u0430\u0435\u0442 \u0434\u0438\u0430\u043f\u0430\u0437\u043e\u043d, \u043a\u043e\u0442\u043e\u0440\u044b\u0439 \u0438\u043d\u0430\u0447\u0435 \u0440\u0430\u0441\u0442\u044f\u043d\u0443\u043b\u0441\u044f \u0431\u044b \u043d\u0430 \u043c\u043d\u043e\u0433\u043e \u043f\u043e\u0440\u044f\u0434\u043a\u043e\u0432, \u0438 \u043f\u043e\u0442\u043e\u043c\u0443 \u0443\u0434\u043e\u0431\u043d\u0430 \u0434\u043b\u044f \u0432\u043e\u0441\u043f\u0440\u0438\u044f\u0442\u0438\u044f \u0438 \u0433\u0440\u0430\u0444\u0438\u043a\u043e\u0432.<\/p>\n<h2 id=\"s6\">\u0412\u044b\u0432\u043e\u0434\u044b<\/h2>\n<ul>\n<li>\u041b\u043e\u0433\u0430\u0440\u0438\u0444\u043c log_a(b) &#8212; \u044d\u0442\u043e \u043f\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c \u0441\u0442\u0435\u043f\u0435\u043d\u0438: \u0441\u0442\u0435\u043f\u0435\u043d\u044c, \u0432 \u043a\u043e\u0442\u043e\u0440\u0443\u044e \u0432\u043e\u0437\u0432\u043e\u0434\u044f\u0442 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435 a, \u0447\u0442\u043e\u0431\u044b \u043f\u043e\u043b\u0443\u0447\u0438\u0442\u044c b; \u043e\u0431\u043b\u0430\u0441\u0442\u044c \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u0437\u0430\u0434\u0430\u044e\u0442 \u0443\u0441\u043b\u043e\u0432\u0438\u044f a &gt; 0, a \u043d\u0435 \u0440\u0430\u0432\u043d\u043e 1, b &gt; 0.<\/li>\n<li>\u0422\u0440\u0438 \u0440\u0430\u0431\u043e\u0447\u0438\u0445 \u0432\u0438\u0434\u0430: \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0439 (\u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435 e, \u0437\u0430\u043f\u0438\u0441\u044c ln), \u0434\u0435\u0441\u044f\u0442\u0438\u0447\u043d\u044b\u0439 (\u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435 10, lg) \u0438 \u0434\u0432\u043e\u0438\u0447\u043d\u044b\u0439 (\u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435 2, log2); \u0432\u0438\u0434 \u0432\u044b\u0431\u0438\u0440\u0430\u044e\u0442 \u043f\u043e\u0434 \u0437\u0430\u0434\u0430\u0447\u0443.<\/li>\n<li>\u0421\u0432\u043e\u0439\u0441\u0442\u0432\u0430 (\u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f, \u0447\u0430\u0441\u0442\u043d\u043e\u0433\u043e, \u0441\u0442\u0435\u043f\u0435\u043d\u0438, \u043f\u0435\u0440\u0435\u0445\u043e\u0434 \u043a \u043d\u043e\u0432\u043e\u043c\u0443 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u044e) \u043f\u0440\u0435\u0432\u0440\u0430\u0449\u0430\u044e\u0442 \u0443\u043c\u043d\u043e\u0436\u0435\u043d\u0438\u0435 \u0438 \u0441\u0442\u0435\u043f\u0435\u043d\u0438 \u0432 \u0441\u043b\u043e\u0436\u0435\u043d\u0438\u0435 &#8212; \u044d\u0442\u043e \u0438 \u0434\u0435\u043b\u0430\u0435\u0442 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \u0443\u0434\u043e\u0431\u043d\u044b\u043c \u0438\u043d\u0441\u0442\u0440\u0443\u043c\u0435\u043d\u0442\u043e\u043c.<\/li>\n<li>\u0412 Python \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u0439\u0442\u0435 <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">math.log2<\/code> \u0438 <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">math.log10<\/code> \u0432\u043c\u0435\u0441\u0442\u043e <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">math.log(x, base)<\/code> \u0434\u043b\u044f \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0439 2 \u0438 10: \u043e\u043d\u0438 \u0442\u043e\u0447\u043d\u0435\u0435 \u0438 \u043d\u0435 \u043a\u043e\u043f\u044f\u0442 \u043e\u0448\u0438\u0431\u043a\u0443 float.<\/li>\n<li>\u0412 IT \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \u0437\u0430\u0434\u0430\u0435\u0442 \u0441\u043b\u043e\u0436\u043d\u043e\u0441\u0442\u044c O(log n), \u043c\u0435\u0440\u0443 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u0438 \u0432 \u0431\u0438\u0442\u0430\u0445 \u0447\u0435\u0440\u0435\u0437 \u044d\u043d\u0442\u0440\u043e\u043f\u0438\u044e \u0438 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0438\u0447\u0435\u0441\u043a\u0438\u0435 \u0448\u043a\u0430\u043b\u044b \u0432\u0440\u043e\u0434\u0435 \u0434\u0435\u0446\u0438\u0431\u0435\u043b.<\/li>\n<\/ul>\n<h2 id=\"s7\">\u0413\u0434\u0435 \u043f\u0440\u0438\u043c\u0435\u043d\u044f\u0435\u0442\u0441\u044f \/ \u0441\u0432\u044f\u0437\u044c \u0441 \u043f\u0440\u0430\u043a\u0442\u0438\u043a\u043e\u0439<\/h2>\n<p>\u041b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u044b &#8212; \u043d\u0435 \u0430\u0431\u0441\u0442\u0440\u0430\u043a\u0446\u0438\u044f \u0438\u0437 \u0443\u0447\u0435\u0431\u043d\u0438\u043a\u0430, \u0430 \u0440\u0430\u0431\u043e\u0447\u0438\u0439 \u0438\u043d\u0441\u0442\u0440\u0443\u043c\u0435\u043d\u0442 \u0440\u0430\u0437\u0440\u0430\u0431\u043e\u0442\u0447\u0438\u043a\u0430 \u0438 \u0430\u043d\u0430\u043b\u0438\u0442\u0438\u043a\u0430. \u041e\u0446\u0435\u043d\u043a\u0430 \u0441\u043b\u043e\u0436\u043d\u043e\u0441\u0442\u0438 \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c\u043e\u0432, \u044d\u043d\u0442\u0440\u043e\u043f\u0438\u044f \u0432 \u0437\u0430\u0434\u0430\u0447\u0430\u0445 \u0441\u0436\u0430\u0442\u0438\u044f \u0438 \u043c\u0430\u0448\u0438\u043d\u043d\u043e\u0433\u043e \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u044f, \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0438\u0447\u0435\u0441\u043a\u0438\u0435 \u0448\u043a\u0430\u043b\u044b \u0432 \u043c\u0435\u0442\u0440\u0438\u043a\u0430\u0445 \u0438 \u043c\u043e\u043d\u0438\u0442\u043e\u0440\u0438\u043d\u0433\u0435 &#8212; \u0432\u0441\u0435 \u044d\u0442\u043e \u043e\u043f\u0438\u0440\u0430\u0435\u0442\u0441\u044f \u043d\u0430 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u0438\u0437 \u0442\u0430\u0431\u043b\u0438\u0446\u044b \u0432\u044b\u0448\u0435. \u041f\u043e\u043d\u0438\u043c\u0430\u043d\u0438\u0435, \u043f\u043e\u0447\u0435\u043c\u0443 \u0431\u0438\u043d\u0430\u0440\u043d\u044b\u0439 \u043f\u043e\u0438\u0441\u043a \u0434\u0430\u0435\u0442 O(log n), \u0430 \u043d\u0435 O(n), \u043d\u0430\u043f\u0440\u044f\u043c\u0443\u044e \u0432\u043b\u0438\u044f\u0435\u0442 \u043d\u0430 \u0432\u044b\u0431\u043e\u0440 \u0441\u0442\u0440\u0443\u043a\u0442\u0443\u0440 \u0434\u0430\u043d\u043d\u044b\u0445.<\/p>\n<div class=\"oj-cta\" style=\"background:#fbf4e4;border-left:4px solid #c9962b;border-radius:8px;padding:16px 20px;margin:24px 0\">\n<p style=\"margin:0 0 8px\"><strong>\u041e\u0441\u0432\u043e\u0439\u0442\u0435 \u0442\u0435\u043c\u0443 \u043d\u0430 \u043f\u0440\u0430\u043a\u0442\u0438\u043a\u0435<\/strong><\/p>\n<p>\u0421\u0438\u0441\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0438 \u0440\u0430\u0437\u043e\u0431\u0440\u0430\u0442\u044c \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u044b, \u043f\u0440\u0435\u0434\u0435\u043b\u044b, \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435 \u0438 \u0434\u0440\u0443\u0433\u0438\u0435 \u0442\u0435\u043c\u044b \u043f\u0440\u0438\u043a\u043b\u0430\u0434\u043d\u043e\u0439 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0438 \u0434\u043b\u044f IT \u043f\u043e\u043c\u043e\u0433\u0430\u0435\u0442 \u043a\u0443\u0440\u0441 <a href=\"https:\/\/otus.ru\/lessons\/prikladnaya-matematika\/?int_article=logarifmy-i-ih-osnovnye-svojstva-chto-nuzhno-znat-specialistu&amp;int_place=article&amp;int_variant=v1\">\u041f\u0440\u0438\u043a\u043b\u0430\u0434\u043d\u0430\u044f \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0432 Otus<\/a>. \u0427\u0442\u043e\u0431\u044b \u043e\u0446\u0435\u043d\u0438\u0442\u044c \u0444\u043e\u0440\u043c\u0430\u0442 \u0438 \u0443\u0440\u043e\u0432\u0435\u043d\u044c \u0434\u043e \u0441\u0442\u0430\u0440\u0442\u0430, \u043f\u043e\u0441\u043c\u043e\u0442\u0440\u0438\u0442\u0435 \u0437\u0430\u043f\u0438\u0441\u0438 \u0438 \u0440\u0430\u0441\u043f\u0438\u0441\u0430\u043d\u0438\u0435 <a href=\"https:\/\/otus.ru\/events\/?int_article=logarifmy-i-ih-osnovnye-svojstva-chto-nuzhno-znat-specialistu&amp;int_place=article&amp;int_variant=v1\">\u043e\u0442\u043a\u0440\u044b\u0442\u044b\u0445 \u0443\u0440\u043e\u043a\u043e\u0432 Otus<\/a> &#8212; \u0442\u0430\u043c \u0440\u0430\u0437\u0431\u0438\u0440\u0430\u044e\u0442 \u043a\u043e\u043d\u043a\u0440\u0435\u0442\u043d\u044b\u0435 \u0437\u0430\u0434\u0430\u0447\u0438 \u0432 \u043f\u0440\u044f\u043c\u043e\u043c \u044d\u0444\u0438\u0440\u0435.<\/p>\n<\/div>\n<h2 id=\"s8\">FAQ<\/h2>\n<p><strong>\u0427\u0435\u043c \u043e\u0442\u043b\u0438\u0447\u0430\u044e\u0442\u0441\u044f ln, lg \u0438 log2?<\/strong><br \/>\n\u042d\u0442\u043e \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u044b \u0441 \u0440\u0430\u0437\u043d\u044b\u043c\u0438 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u044f\u043c\u0438: ln &#8212; \u043f\u043e \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u044e e (\u043f\u0440\u0438\u043c\u0435\u0440\u043d\u043e 2.71828), lg &#8212; \u043f\u043e \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u044e 10, log2 &#8212; \u043f\u043e \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u044e 2. \u0424\u043e\u0440\u043c\u0443\u043b\u0430 \u043f\u0435\u0440\u0435\u0445\u043e\u0434\u0430 \u0441\u0432\u044f\u0437\u044b\u0432\u0430\u0435\u0442 \u0438\u0445: \u043b\u044e\u0431\u043e\u0439 \u0438\u0437 \u043d\u0438\u0445 \u0432\u044b\u0440\u0430\u0436\u0430\u0435\u0442\u0441\u044f \u0447\u0435\u0440\u0435\u0437 \u0434\u0440\u0443\u0433\u043e\u0439 \u0434\u0435\u043b\u0435\u043d\u0438\u0435\u043c, \u043d\u0430\u043f\u0440\u0438\u043c\u0435\u0440 log2(x) = ln(x) \/ ln(2).<\/p>\n<p><strong>\u041f\u043e\u0447\u0435\u043c\u0443 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0430 \u043d\u0435 \u043c\u043e\u0436\u0435\u0442 \u0431\u044b\u0442\u044c 1 \u0438\u043b\u0438 \u043e\u0442\u0440\u0438\u0446\u0430\u0442\u0435\u043b\u044c\u043d\u044b\u043c?<\/strong><br \/>\n\u0415\u0434\u0438\u043d\u0438\u0446\u0430 \u0432 \u043b\u044e\u0431\u043e\u0439 \u0441\u0442\u0435\u043f\u0435\u043d\u0438 \u0440\u0430\u0432\u043d\u0430 \u0435\u0434\u0438\u043d\u0438\u0446\u0435, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u0435 1^x = b \u043d\u0435 \u0438\u043c\u0435\u0435\u0442 \u0440\u0435\u0448\u0435\u043d\u0438\u044f \u043f\u0440\u0438 b, \u043e\u0442\u043b\u0438\u0447\u043d\u043e\u043c \u043e\u0442 1. \u041e\u0442\u0440\u0438\u0446\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0435 \u0438\u043b\u0438 \u043d\u0443\u043b\u0435\u0432\u043e\u0435 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435 \u0434\u0430\u0435\u0442 \u0441\u0442\u0435\u043f\u0435\u043d\u0438, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043d\u0435 \u043f\u043e\u043a\u0440\u044b\u0432\u0430\u044e\u0442 \u043f\u043e\u043b\u043e\u0436\u0438\u0442\u0435\u043b\u044c\u043d\u044b\u0435 \u0447\u0438\u0441\u043b\u0430 \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u043e, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \u043f\u043e \u0442\u0430\u043a\u043e\u043c\u0443 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u044e \u0432 \u0432\u0435\u0449\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u0445 \u0447\u0438\u0441\u043b\u0430\u0445 \u043d\u0435 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u044f\u044e\u0442.<\/p>\n<p><strong>\u041c\u043e\u0436\u043d\u043e \u043b\u0438 \u0432\u0437\u044f\u0442\u044c \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \u043e\u0442 \u043d\u0443\u043b\u044f \u0438\u043b\u0438 \u043e\u0442\u0440\u0438\u0446\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0433\u043e \u0447\u0438\u0441\u043b\u0430?<\/strong><br \/>\n\u0412 \u0432\u0435\u0449\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u0445 \u0447\u0438\u0441\u043b\u0430\u0445 &#8212; \u043d\u0435\u0442: \u0430\u0440\u0433\u0443\u043c\u0435\u043d\u0442 \u0434\u043e\u043b\u0436\u0435\u043d \u0431\u044b\u0442\u044c \u0441\u0442\u0440\u043e\u0433\u043e \u0431\u043e\u043b\u044c\u0448\u0435 \u043d\u0443\u043b\u044f, \u0442\u0430\u043a \u043a\u0430\u043a \u043f\u043e\u043b\u043e\u0436\u0438\u0442\u0435\u043b\u044c\u043d\u043e\u0435 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435 \u0432 \u043b\u044e\u0431\u043e\u0439 \u0432\u0435\u0449\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u043e\u0439 \u0441\u0442\u0435\u043f\u0435\u043d\u0438 \u043f\u043e\u043b\u043e\u0436\u0438\u0442\u0435\u043b\u044c\u043d\u043e. \u0414\u043b\u044f \u043e\u0442\u0440\u0438\u0446\u0430\u0442\u0435\u043b\u044c\u043d\u044b\u0445 \u0438 \u043d\u0443\u043b\u0435\u0432\u044b\u0445 \u0430\u0440\u0433\u0443\u043c\u0435\u043d\u0442\u043e\u0432 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u044f\u044e\u0442 \u0442\u043e\u043b\u044c\u043a\u043e \u0432 \u043a\u043e\u043c\u043f\u043b\u0435\u043a\u0441\u043d\u043e\u043c \u0430\u043d\u0430\u043b\u0438\u0437\u0435, \u0447\u0442\u043e \u0432\u044b\u0445\u043e\u0434\u0438\u0442 \u0437\u0430 \u0440\u0430\u043c\u043a\u0438 \u0442\u0438\u043f\u043e\u0432\u044b\u0445 \u043f\u0440\u0438\u043a\u043b\u0430\u0434\u043d\u044b\u0445 \u0437\u0430\u0434\u0430\u0447.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u041b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \u0447\u0438\u0441\u043b\u0430 b \u043f\u043e \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u044e a &#8212; \u044d\u0442\u043e \u043f\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c \u0441\u0442\u0435\u043f\u0435\u043d\u0438, \u0432 \u043a\u043e\u0442\u043e\u0440\u0443\u044e \u043d\u0443\u0436\u043d\u043e \u0432\u043e\u0437\u0432\u0435\u0441\u0442\u0438 a, \u0447\u0442\u043e\u0431\u044b \u043f\u043e\u043b\u0443\u0447\u0438\u0442\u044c b; \u0437\u0430\u043f\u0438\u0441\u044b\u0432\u0430\u0435\u0442\u0441\u044f \u043a\u0430\u043a log_a(b). \u0418\u043d\u0430\u0447\u0435 \u0433\u043e\u0432\u043e\u0440\u044f, \u0435\u0441\u043b\u0438 a^x = b, \u0442\u043e x = log_a(b). \u041b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u0435 &#8212; \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u044f, \u043e\u0431\u0440\u0430\u0442\u043d\u0430\u044f \u0432\u043e\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044e \u0432 \u0441\u0442\u0435\u043f\u0435\u043d\u044c: \u043e\u043d\u0430 \u043e\u0442\u0432\u0435\u0447\u0430\u0435\u0442 \u043d\u0430 \u0432\u043e\u043f\u0440\u043e\u0441 \u00ab\u0432 \u043a\u0430\u043a\u0443\u044e \u0441\u0442\u0435\u043f\u0435\u043d\u044c \u0432\u043e\u0437\u0432\u0435\u0441\u0442\u0438 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435, \u0447\u0442\u043e\u0431\u044b \u043f\u043e\u043b\u0443\u0447\u0438\u0442\u044c \u0447\u0438\u0441\u043b\u043e\u00bb. \u0421\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435 \u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0438 \u0443\u0441\u043b\u043e\u0432\u0438\u044f \u0422\u0440\u0438 [&hellip;]<\/p>\n","protected":false},"author":11,"featured_media":11226,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[27,558,245,65],"class_list":["post-11225","post","type-post","status-publish","format-standard","has-post-thumbnail","","category-polza","tag-python","tag-logarifmy","tag-matematika-2","tag-programmirovanie"],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO 5.0.3 - aioseo.com -->\n\t<meta name=\"description\" content=\"\u041b\u043e\u0433\u0430\u0440\u0438\u0444\u043c - \u044d\u0442\u043e \u043f\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c \u0441\u0442\u0435\u043f\u0435\u043d\u0438. \u0420\u0430\u0437\u0431\u0438\u0440\u0430\u0435\u043c \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435, \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u044f (ln, lg, log2), \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430 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Pavlenko\"\/>\n\t<link rel=\"canonical\" href=\"https:\/\/otus.ru\/journal\/logarifmy-i-ih-osnovnye-svojstva-chto-nuzhno-znat-specialistu\/\" \/>\n\t\t<meta name=\"generator\" content=\"All in One SEO (AIOSEO) 5.0.3\" \/>\n\t\t<script type=\"application\/ld+json\" class=\"aioseo-schema\">\n\t\t\t{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/otus.ru\\\/journal\\\/logarifmy-i-ih-osnovnye-svojstva-chto-nuzhno-znat-specialistu\\\/#article\",\"name\":\"\\u041b\\u043e\\u0433\\u0430\\u0440\\u0438\\u0444\\u043c\\u044b: \\u0441\\u0432\\u043e\\u0439\\u0441\\u0442\\u0432\\u0430 \\u0438 \\u043f\\u0440\\u0438\\u043c\\u0435\\u043d\\u0435\\u043d\\u0438\\u0435 \\u0432 IT | Otus\",\"headline\":\"\\u041b\\u043e\\u0433\\u0430\\u0440\\u0438\\u0444\\u043c\\u044b \\u0438 \\u0438\\u0445 \\u043e\\u0441\\u043d\\u043e\\u0432\\u043d\\u044b\\u0435 \\u0441\\u0432\\u043e\\u0439\\u0441\\u0442\\u0432\\u0430: \\u0447\\u0442\\u043e 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