{"id":11219,"date":"2026-09-18T04:22:29","date_gmt":"2026-09-18T04:22:29","guid":{"rendered":"https:\/\/otus.ru\/journal\/?p=11219"},"modified":"2026-09-25T15:46:31","modified_gmt":"2026-09-25T15:46:31","slug":"logarifmy-opisanie-i-graficheskoe-izobrazhenie","status":"publish","type":"post","link":"https:\/\/otus.ru\/journal\/logarifmy-opisanie-i-graficheskoe-izobrazhenie\/","title":{"rendered":"\u041b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u044b: \u043e\u043f\u0438\u0441\u0430\u043d\u0438\u0435, \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u0438 \u0433\u0440\u0430\u0444\u0438\u043a \u0444\u0443\u043d\u043a\u0446\u0438\u0438"},"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 c, \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: \u0435\u0441\u043b\u0438 a^c = b, \u0442\u043e log_a(b) = c. \u041f\u0440\u043e\u0449\u0435 \u0433\u043e\u0432\u043e\u0440\u044f, \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \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 \u0430\u0440\u0433\u0443\u043c\u0435\u043d\u0442\u00bb. \u041d\u0438\u0436\u0435 \u0440\u0430\u0437\u0431\u0435\u0440\u0435\u043c \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0441 \u0443\u0441\u043b\u043e\u0432\u0438\u044f\u043c\u0438, \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u0438 \u0444\u043e\u0440\u043c\u0443\u043b\u044b, \u0442\u0440\u0438 \u0445\u043e\u0434\u043e\u0432\u044b\u0445 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u044f (2, e, 10), \u0433\u0440\u0430\u0444\u0438\u043a \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0438 \u0442\u043e, \u043e\u0442\u043a\u0443\u0434\u0430 \u0432 IT \u0431\u0435\u0440\u0435\u0442\u0441\u044f \u0441\u043b\u043e\u0436\u043d\u043e\u0441\u0442\u044c O(log n).<\/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 \u043e\u0431\u043e\u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435<\/a><\/li>\n<li><a href=\"#s2\">\u0422\u0440\u0438 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u044f: 2, e \u0438 10<\/a><\/li>\n<li><a href=\"#s3\">\u0421\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u0438 \u0444\u043e\u0440\u043c\u0443\u043b\u044b<\/a><\/li>\n<li><a href=\"#s4\">\u041b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0438\u0447\u0435\u0441\u043a\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0438 \u0435\u0435 \u0433\u0440\u0430\u0444\u0438\u043a<\/a><\/li>\n<li><a href=\"#s5\">\u041e\u0442\u043a\u0443\u0434\u0430 \u0432 IT \u0431\u0435\u0440\u0435\u0442\u0441\u044f O(log n)<\/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>\u0421\u0440\u0430\u0437\u0443 \u0440\u0430\u0437\u0432\u0435\u0434\u0435\u043c \u0442\u0440\u0438 \u0431\u043b\u0438\u0437\u043a\u0438\u0445 \u043f\u043e\u043d\u044f\u0442\u0438\u044f, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043b\u0435\u0433\u043a\u043e \u0441\u043f\u0443\u0442\u0430\u0442\u044c: <strong>\u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c<\/strong> &#8212; \u044d\u0442\u043e \u0447\u0438\u0441\u043b\u043e (\u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442); <strong>\u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u0435<\/strong> &#8212; \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u044f \u043d\u0430\u0445\u043e\u0436\u0434\u0435\u043d\u0438\u044f \u044d\u0442\u043e\u0433\u043e \u0447\u0438\u0441\u043b\u0430; <strong>\u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0438\u0447\u0435\u0441\u043a\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f<\/strong> y = log_a(x) &#8212; \u044d\u0442\u043e \u043f\u0440\u0430\u0432\u0438\u043b\u043e, \u043a\u043e\u0442\u043e\u0440\u043e\u0435 \u043a\u0430\u0436\u0434\u043e\u043c\u0443 x \u0441\u0442\u0430\u0432\u0438\u0442 \u0432 \u0441\u043e\u043e\u0442\u0432\u0435\u0442\u0441\u0442\u0432\u0438\u0435 \u0435\u0433\u043e \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c. \u0414\u0430\u043b\u044c\u0448\u0435 \u043f\u043e \u0442\u0435\u043a\u0441\u0442\u0443 \u0434\u0435\u0440\u0436\u0438\u043c \u044d\u0442\u043e \u0440\u0430\u0437\u043b\u0438\u0447\u0438\u0435.<\/p>\n<h2 id=\"s1\">\u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0438 \u043e\u0431\u043e\u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435<\/h2>\n<p>\u0417\u0430\u043f\u0438\u0441\u044c log_a(b) \u0447\u0438\u0442\u0430\u0435\u0442\u0441\u044f \u043a\u0430\u043a \u00ab\u043b\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\u00bb. \u0417\u0434\u0435\u0441\u044c b &#8212; \u0430\u0440\u0433\u0443\u043c\u0435\u043d\u0442, a &#8212; \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435. \u041b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u0435 &#8212; \u044d\u0442\u043e \u043e\u0434\u043d\u0430 \u0438\u0437 \u0434\u0432\u0443\u0445 \u043e\u0431\u0440\u0430\u0442\u043d\u044b\u0445 \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u0439 \u043a \u0432\u043e\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044e \u0432 \u0441\u0442\u0435\u043f\u0435\u043d\u044c: \u0435\u0441\u043b\u0438 \u043f\u043e a^c = b \u0438\u0441\u043a\u0430\u0442\u044c \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435 a, \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c \u043a\u043e\u0440\u0435\u043d\u044c; \u0435\u0441\u043b\u0438 \u0438\u0441\u043a\u0430\u0442\u044c \u043f\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c c &#8212; \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c.<\/p>\n<p>\u041e\u0433\u0440\u0430\u043d\u0438\u0447\u0435\u043d\u0438\u044f (\u0431\u0435\u0437 \u043d\u0438\u0445 \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0435 \u043d\u0435 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u043e): \u0430\u0440\u0433\u0443\u043c\u0435\u043d\u0442 b &gt; 0, \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435 a &gt; 0 \u0438 a \u2260 1. \u041f\u0440\u0438\u0447\u0438\u043d\u0430 \u043f\u0440\u043e\u0441\u0442\u0430: \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 \u0432\u0441\u0435\u0433\u0434\u0430 \u043f\u043e\u043b\u043e\u0436\u0438\u0442\u0435\u043b\u044c\u043d\u0430, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u043e\u0442\u0440\u0438\u0446\u0430\u0442\u0435\u043b\u044c\u043d\u044b\u0435 \u0447\u0438\u0441\u043b\u0430 \u0438 \u043d\u043e\u043b\u044c \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0438\u0440\u043e\u0432\u0430\u0442\u044c \u043d\u0435\u043b\u044c\u0437\u044f, \u0430 \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, \u0438 \u043e\u0434\u043d\u043e\u0437\u043d\u0430\u0447\u043d\u043e \u00ab\u043f\u043e\u0434\u043e\u0431\u0440\u0430\u0442\u044c \u043f\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u00bb \u0434\u043b\u044f \u0434\u0440\u0443\u0433\u043e\u0433\u043e \u0430\u0440\u0433\u0443\u043c\u0435\u043d\u0442\u0430 \u043d\u0435 \u0432\u044b\u0439\u0434\u0435\u0442.<\/p>\n<p>\u0411\u044b\u0441\u0442\u0440\u0430\u044f \u043f\u0440\u043e\u0432\u0435\u0440\u043a\u0430 \u043f\u043e\u043d\u0438\u043c\u0430\u043d\u0438\u044f: log_2(8) = 3, \u043f\u043e\u0442\u043e\u043c\u0443 \u0447\u0442\u043e 2^3 = 8; log_10(1000) = 3, \u043f\u043e\u0442\u043e\u043c\u0443 \u0447\u0442\u043e 10^3 = 1000.<\/p>\n<h2 id=\"s2\">\u0422\u0440\u0438 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u044f: 2, e \u0438 10<\/h2>\n<p>\u041e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435 \u043c\u043e\u0436\u0435\u0442 \u0431\u044b\u0442\u044c \u043b\u044e\u0431\u044b\u043c \u0434\u043e\u043f\u0443\u0441\u0442\u0438\u043c\u044b\u043c \u0447\u0438\u0441\u043b\u043e\u043c, \u043d\u043e \u043d\u0430 \u043f\u0440\u0430\u043a\u0442\u0438\u043a\u0435 \u0432\u0441\u0442\u0440\u0435\u0447\u0430\u044e\u0442\u0441\u044f \u0442\u0440\u0438. \u0423 \u043a\u0430\u0436\u0434\u043e\u0433\u043e &#8212; \u0441\u0432\u043e\u044f \u043a\u043e\u0440\u043e\u0442\u043a\u0430\u044f \u0437\u0430\u043f\u0438\u0441\u044c, \u0438\u0445 \u0432\u0430\u0436\u043d\u043e \u043d\u0435 \u043f\u0443\u0442\u0430\u0442\u044c.<\/p>\n<table>\n<thead>\n<tr>\n<th>\u041e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435<\/th>\n<th>\u041d\u0430\u0437\u0432\u0430\u043d\u0438\u0435<\/th>\n<th>\u0417\u0430\u043f\u0438\u0441\u044c<\/th>\n<th>\u0413\u0434\u0435 \u043f\u0440\u0438\u043c\u0435\u043d\u044f\u0435\u0442\u0441\u044f<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>2<\/td>\n<td>\u0434\u0432\u043e\u0438\u0447\u043d\u044b\u0439<\/td>\n<td>log_2(x), lb(x), \u0432 CS \u0447\u0430\u0441\u0442\u043e \u043f\u0440\u043e\u0441\u0442\u043e log(x)<\/td>\n<td>\u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c\u044b, \u0441\u0442\u0440\u0443\u043a\u0442\u0443\u0440\u044b \u0434\u0430\u043d\u043d\u044b\u0445, \u043e\u0446\u0435\u043d\u043a\u0430 \u0441\u043b\u043e\u0436\u043d\u043e\u0441\u0442\u0438<\/td>\n<\/tr>\n<tr>\n<td>e \u2248 2,71828<\/td>\n<td>\u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0439<\/td>\n<td>ln(x)<\/td>\n<td>\u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u044b\u0439 \u0440\u043e\u0441\u0442, \u043c\u0430\u0442\u0430\u043d\u0430\u043b\u0438\u0437, \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0435<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>\u0434\u0435\u0441\u044f\u0442\u0438\u0447\u043d\u044b\u0439<\/td>\n<td>lg(x), log_10(x)<\/td>\n<td>\u0438\u043d\u0436\u0435\u043d\u0435\u0440\u043d\u044b\u0435 \u0448\u043a\u0430\u043b\u044b (\u0434\u0411, pH), \u00ab\u043f\u043e\u0440\u044f\u0434\u043a\u0438 \u0432\u0435\u043b\u0438\u0447\u0438\u043d\u00bb<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u0427\u0438\u0441\u043b\u043e e &#8212; \u044d\u0442\u043e \u0438\u0440\u0440\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u044c\u043d\u0430\u044f \u043a\u043e\u043d\u0441\u0442\u0430\u043d\u0442\u0430 (\u0435\u0435 \u043d\u0435\u043b\u044c\u0437\u044f \u0437\u0430\u043f\u0438\u0441\u0430\u0442\u044c \u043a\u043e\u043d\u0435\u0447\u043d\u043e\u0439 \u0438\u043b\u0438 \u043f\u0435\u0440\u0438\u043e\u0434\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u0434\u0440\u043e\u0431\u044c\u044e), \u043f\u0440\u0438\u0431\u043b\u0438\u0437\u0438\u0442\u0435\u043b\u044c\u043d\u043e 2,71828. \u041e\u043d\u0430 \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 \u0441\u0443\u043c\u043c\u0430 \u0440\u0430\u0441\u0442\u0435\u0442 \u043d\u0430 100% \u0432 \u0433\u043e\u0434, \u043d\u043e \u043d\u0430\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u0435 \u0438\u0434\u0435\u0442 \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u043e, \u0430 \u043d\u0435 \u0440\u0430\u0437\u043e\u043c \u0432 \u043a\u043e\u043d\u0446\u0435 \u0433\u043e\u0434\u0430, \u0437\u0430 \u0433\u043e\u0434 \u043e\u043d\u0430 \u0443\u043c\u043d\u043e\u0436\u0430\u0435\u0442\u0441\u044f \u043d\u0435 \u043d\u0430 2, \u0430 \u043d\u0430 e. \u0418\u043c\u0435\u043d\u043d\u043e \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0439 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \u0443\u0434\u043e\u0431\u0435\u043d \u0434\u043b\u044f \u043e\u043f\u0438\u0441\u0430\u043d\u0438\u044f \u043f\u0440\u043e\u0446\u0435\u0441\u0441\u043e\u0432 \u0432\u0438\u0434\u0430 \u00ab\u0441\u043a\u043e\u0440\u043e\u0441\u0442\u044c \u0440\u043e\u0441\u0442\u0430 \u043f\u0440\u043e\u043f\u043e\u0440\u0446\u0438\u043e\u043d\u0430\u043b\u044c\u043d\u0430 \u0442\u0435\u043a\u0443\u0449\u0435\u043c\u0443 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044e\u00bb &#8212; \u0440\u0430\u0434\u0438\u043e\u0430\u043a\u0442\u0438\u0432\u043d\u043e\u0433\u043e \u0440\u0430\u0441\u043f\u0430\u0434\u0430, \u043e\u0441\u0442\u044b\u0432\u0430\u043d\u0438\u044f, \u043d\u0430\u043a\u043e\u043f\u043b\u0435\u043d\u0438\u044f \u043f\u0440\u043e\u0446\u0435\u043d\u0442\u043e\u0432.<\/p>\n<p>\u041e\u0433\u043e\u0432\u043e\u0440\u043a\u0430 \u043f\u0440\u043e \u0437\u0430\u043f\u0438\u0441\u044c: \u043e\u0431\u043e\u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435 lb \u0434\u043b\u044f \u0434\u0432\u043e\u0438\u0447\u043d\u043e\u0433\u043e \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0430 \u0441\u0442\u0430\u043d\u0434\u0430\u0440\u0442\u0438\u0437\u0438\u0440\u043e\u0432\u0430\u043d\u043e (ISO 80000-2), \u043d\u043e \u0432 \u0442\u0435\u043a\u0441\u0442\u0430\u0445 \u043f\u043e \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c\u0430\u043c \u0447\u0430\u0449\u0435 \u043f\u0438\u0448\u0443\u0442 \u043f\u0440\u043e\u0441\u0442\u043e log n, \u043f\u043e\u0434\u0440\u0430\u0437\u0443\u043c\u0435\u0432\u0430\u044f \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435 2. \u042d\u0442\u043e \u043d\u0435 \u043d\u0435\u0431\u0440\u0435\u0436\u043d\u043e\u0441\u0442\u044c: \u043f\u0440\u0438 \u043e\u0446\u0435\u043d\u043a\u0435 \u0430\u0441\u0438\u043c\u043f\u0442\u043e\u0442\u0438\u043a\u0438 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435 \u0432\u043b\u0438\u044f\u0435\u0442 \u043b\u0438\u0448\u044c \u043d\u0430 \u043f\u043e\u0441\u0442\u043e\u044f\u043d\u043d\u044b\u0439 \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044c (\u0441\u043c. \u0444\u043e\u0440\u043c\u0443\u043b\u0443 \u043f\u0435\u0440\u0435\u0445\u043e\u0434\u0430 \u043d\u0438\u0436\u0435), \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u0432 \u0437\u0430\u043f\u0438\u0441\u0438 O(log n) \u0435\u0433\u043e \u043e\u043f\u0443\u0441\u043a\u0430\u044e\u0442.<\/p>\n<h2 id=\"s3\">\u0421\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u0438 \u0444\u043e\u0440\u043c\u0443\u043b\u044b<\/h2>\n<p>\u0412\u0441\u0435 \u0444\u043e\u0440\u043c\u0443\u043b\u044b \u043d\u0438\u0436\u0435 \u0440\u0430\u0431\u043e\u0442\u0430\u044e\u0442 \u0442\u043e\u043b\u044c\u043a\u043e \u043f\u0440\u0438 \u0443\u0441\u043b\u043e\u0432\u0438\u0438, \u0447\u0442\u043e \u0430\u0440\u0433\u0443\u043c\u0435\u043d\u0442\u044b \u043f\u043e\u043b\u043e\u0436\u0438\u0442\u0435\u043b\u044c\u043d\u044b, \u0430 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435 \u0434\u043e\u043f\u0443\u0441\u0442\u0438\u043c\u043e (a &gt; 0, a \u2260 1). \u042d\u0442\u043e \u043d\u0435 \u0443\u043a\u0440\u0430\u0448\u0435\u043d\u0438\u0435, \u0430 \u0447\u0430\u0441\u0442\u044c \u043f\u0440\u0430\u0432\u0438\u043b\u0430.<\/p>\n<ul>\n<li>log_a(1) = 0 &#8212; \u0435\u0434\u0438\u043d\u0438\u0446\u0430 \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u0442\u0441\u044f \u0432\u043e\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435\u043c \u0432 \u043d\u0443\u043b\u0435\u0432\u0443\u044e \u0441\u0442\u0435\u043f\u0435\u043d\u044c.<\/li>\n<li>log_a(a) = 1 &#8212; \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435 \u0432 \u043f\u0435\u0440\u0432\u043e\u0439 \u0441\u0442\u0435\u043f\u0435\u043d\u0438 \u0434\u0430\u0435\u0442 \u0441\u0430\u043c\u043e \u0441\u0435\u0431\u044f.<\/li>\n<li>log_a(x \u00b7 y) = log_a(x) + log_a(y) \u043f\u0440\u0438 x &gt; 0, y &gt; 0 &#8212; \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f \u0440\u0430\u0432\u0435\u043d \u0441\u0443\u043c\u043c\u0435 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u043e\u0432.<\/li>\n<li>log_a(x \/ y) = log_a(x) &#8212; log_a(y) \u043f\u0440\u0438 x &gt; 0, y &gt; 0 &#8212; \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \u0447\u0430\u0441\u0442\u043d\u043e\u0433\u043e \u0440\u0430\u0432\u0435\u043d \u0440\u0430\u0437\u043d\u043e\u0441\u0442\u0438.<\/li>\n<li>log_a(x^p) = p \u00b7 log_a(x) \u043f\u0440\u0438 x &gt; 0 &#8212; \u043f\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c \u0441\u0442\u0435\u043f\u0435\u043d\u0438 \u0432\u044b\u043d\u043e\u0441\u0438\u0442\u0441\u044f \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0435\u043c.<\/li>\n<li>\u041f\u0435\u0440\u0435\u0445\u043e\u0434 \u043a \u043d\u043e\u0432\u043e\u043c\u0443 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u044e: log_a(x) = log_c(x) \/ log_c(a). \u041e\u0442\u0441\u044e\u0434\u0430, \u043d\u0430\u043f\u0440\u0438\u043c\u0435\u0440, log_2(x) = ln(x) \/ ln(2) &#8212; \u0438\u043c\u0435\u043d\u043d\u043e \u0442\u0430\u043a \u0434\u0432\u043e\u0438\u0447\u043d\u044b\u0439 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \u0441\u0447\u0438\u0442\u0430\u044e\u0442 \u0447\u0435\u0440\u0435\u0437 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0439.<\/li>\n<\/ul>\n<p>\u041f\u0440\u043e\u0432\u0435\u0440\u0438\u043c \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u043e \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f \u0438 \u043f\u0435\u0440\u0435\u0445\u043e\u0434 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u044f \u043d\u0430 \u0447\u0438\u0441\u043b\u0430\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: #7F848E\"># \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f = \u0441\u0443\u043c\u043c\u0430 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u043e\u0432<\/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\">4<\/span> <span style=\"color: #56B6C2\">*<\/span> <span style=\"color: #D19A66\">8<\/span><span style=\"color: #ABB2BF\">))<\/span>            <span style=\"color: #7F848E\"># 5.0<\/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\">4<\/span><span style=\"color: #ABB2BF\">)<\/span> <span style=\"color: #56B6C2\">+<\/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\"># 5.0<\/span>\n\n<span style=\"color: #7F848E\"># \u043f\u0435\u0440\u0435\u0445\u043e\u0434 \u043a \u043d\u043e\u0432\u043e\u043c\u0443 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u044e: log_3(81) \u0447\u0435\u0440\u0435\u0437 \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0439<\/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\">81<\/span><span style=\"color: #ABB2BF\">)<\/span> <span style=\"color: #56B6C2\">\/<\/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\">3<\/span><span style=\"color: #ABB2BF\">))<\/span>  <span style=\"color: #7F848E\"># 4.0  (3**4 == 81)<\/span>\n\n<span style=\"color: #7F848E\"># math.log(x, base) \u0434\u0435\u043b\u0430\u0435\u0442 \u043f\u0435\u0440\u0435\u0445\u043e\u0434 \u0432\u043d\u0443\u0442\u0440\u0438<\/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\">81<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">3<\/span><span style=\"color: #ABB2BF\">))<\/span>             <span style=\"color: #7F848E\"># 4.0<\/span><\/code><\/pre>\n<p>\u0424\u0430\u043a\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0439 \u0432\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\">5.0<\/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\">5.0<\/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\">4.0<\/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\">4.0<\/code>. \u041e\u0431\u0440\u0430\u0442\u0438\u0442\u0435 \u0432\u043d\u0438\u043c\u0430\u043d\u0438\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.log(x)<\/code> \u0431\u0435\u0437 \u0432\u0442\u043e\u0440\u043e\u0433\u043e \u0430\u0440\u0433\u0443\u043c\u0435\u043d\u0442\u0430 &#8212; \u044d\u0442\u043e \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0439 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c (\u043f\u043e \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u044e e), \u0434\u043b\u044f \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u044f 10 \u0435\u0441\u0442\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\">math.log10<\/code>, \u0434\u043b\u044f 2 &#8212; <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>.<\/p>\n<h2 id=\"s4\">\u041b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0438\u0447\u0435\u0441\u043a\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0438 \u0435\u0435 \u0433\u0440\u0430\u0444\u0438\u043a<\/h2>\n<p>\u041b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0438\u0447\u0435\u0441\u043a\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f &#8212; \u044d\u0442\u043e y = log_a(x) \u043f\u0440\u0438 a &gt; 0 \u0438 a \u2260 1. \u041e\u043d\u0430 \u043e\u0431\u0440\u0430\u0442\u043d\u0430 \u043f\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 y = a^x: \u0438\u0445 \u0433\u0440\u0430\u0444\u0438\u043a\u0438 \u0441\u0438\u043c\u043c\u0435\u0442\u0440\u0438\u0447\u043d\u044b \u043e\u0442\u043d\u043e\u0441\u0438\u0442\u0435\u043b\u044c\u043d\u043e \u043f\u0440\u044f\u043c\u043e\u0439 y = x.<\/p>\n<p>\u0421\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0438 \u0442\u043e, \u043a\u0430\u043a \u043e\u043d\u0438 \u0447\u0438\u0442\u0430\u044e\u0442\u0441\u044f \u043d\u0430 \u0433\u0440\u0430\u0444\u0438\u043a\u0435:<\/p>\n<ul>\n<li>\u041e\u0431\u043b\u0430\u0441\u0442\u044c \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f D(f) = (0; +\u221e) &#8212; \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u0435\u0442 \u0442\u043e\u043b\u044c\u043a\u043e \u0434\u043b\u044f \u043f\u043e\u043b\u043e\u0436\u0438\u0442\u0435\u043b\u044c\u043d\u044b\u0445 x; \u0441\u043b\u0435\u0432\u0430 \u043e\u0442 \u043d\u0443\u043b\u044f \u0433\u0440\u0430\u0444\u0438\u043a\u0430 \u043d\u0435\u0442.<\/li>\n<li>\u041e\u0431\u043b\u0430\u0441\u0442\u044c \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0439 E(f) = (-\u221e; +\u221e) &#8212; \u043f\u043e \u0432\u0435\u0440\u0442\u0438\u043a\u0430\u043b\u0438 \u043a\u0440\u0438\u0432\u0430\u044f \u0443\u0445\u043e\u0434\u0438\u0442 \u0438 \u0432\u043d\u0438\u0437, \u0438 \u0432\u0432\u0435\u0440\u0445 \u0431\u0435\u0437 \u043e\u0433\u0440\u0430\u043d\u0438\u0447\u0435\u043d\u0438\u0439.<\/li>\n<li>\u0413\u0440\u0430\u0444\u0438\u043a \u0432\u0441\u0435\u0433\u0434\u0430 \u043f\u0440\u043e\u0445\u043e\u0434\u0438\u0442 \u0447\u0435\u0440\u0435\u0437 \u0442\u043e\u0447\u043a\u0443 (1; 0), \u043f\u043e\u0442\u043e\u043c\u0443 \u0447\u0442\u043e log_a(1) = 0 \u043f\u0440\u0438 \u043b\u044e\u0431\u043e\u043c \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0438, \u0438 \u0447\u0435\u0440\u0435\u0437 \u0442\u043e\u0447\u043a\u0443 (a; 1).<\/li>\n<li>\u041e\u0441\u044c \u043e\u0440\u0434\u0438\u043d\u0430\u0442 (\u043f\u0440\u044f\u043c\u0430\u044f x = 0) &#8212; \u0432\u0435\u0440\u0442\u0438\u043a\u0430\u043b\u044c\u043d\u0430\u044f \u0430\u0441\u0438\u043c\u043f\u0442\u043e\u0442\u0430: \u043f\u0440\u0438 x, \u0441\u0442\u0440\u0435\u043c\u044f\u0449\u0435\u043c\u0441\u044f \u043a \u043d\u0443\u043b\u044e \u0441\u043f\u0440\u0430\u0432\u0430, \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435 \u0443\u0445\u043e\u0434\u0438\u0442 \u043a -\u221e (\u043f\u0440\u0438 a &gt; 1) \u0438\u043b\u0438 \u043a +\u221e (\u043f\u0440\u0438 0 &lt; a &lt; 1), \u043d\u043e \u0441\u0430\u043c\u0443 \u043e\u0441\u044c \u0433\u0440\u0430\u0444\u0438\u043a \u043d\u0435 \u043f\u0435\u0440\u0435\u0441\u0435\u043a\u0430\u0435\u0442.<\/li>\n<li>\u041f\u0440\u0438 a &gt; 1 \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0441\u0442\u0440\u043e\u0433\u043e \u0432\u043e\u0437\u0440\u0430\u0441\u0442\u0430\u0435\u0442 \u043d\u0430 \u0432\u0441\u0435\u0439 \u043e\u0431\u043b\u0430\u0441\u0442\u0438 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f; \u043f\u0440\u0438 0 &lt; a &lt; 1 &#8212; \u0441\u0442\u0440\u043e\u0433\u043e \u0443\u0431\u044b\u0432\u0430\u0435\u0442.<\/li>\n<li>\u0424\u0443\u043d\u043a\u0446\u0438\u044f \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u0430 \u0438 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0440\u0443\u0435\u043c\u0430 \u0432\u0441\u044e\u0434\u0443, \u0433\u0434\u0435 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0430; \u043e\u043d\u0430 \u043d\u0435 \u0447\u0435\u0442\u043d\u0430\u044f \u0438 \u043d\u0435 \u043d\u0435\u0447\u0435\u0442\u043d\u0430\u044f, \u043d\u0435 \u0438\u043c\u0435\u0435\u0442 \u043d\u0438 \u043d\u0430\u0438\u0431\u043e\u043b\u044c\u0448\u0435\u0433\u043e, \u043d\u0438 \u043d\u0430\u0438\u043c\u0435\u043d\u044c\u0448\u0435\u0433\u043e \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044f.<\/li>\n<\/ul>\n<p>\u041f\u0440\u043e\u0449\u0435 \u0432\u0441\u0435\u0433\u043e \u0443\u0432\u0438\u0434\u0435\u0442\u044c \u0444\u043e\u0440\u043c\u0443 \u043f\u043e \u0442\u0430\u0431\u043b\u0438\u0446\u0435 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0439. \u0414\u043b\u044f y = log_2(x) (\u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435 \u0431\u043e\u043b\u044c\u0448\u0435 1, \u043a\u0440\u0438\u0432\u0430\u044f \u0432\u043e\u0437\u0440\u0430\u0441\u0442\u0430\u0435\u0442):<\/p>\n<table>\n<thead>\n<tr>\n<th>x<\/th>\n<th>1\/4<\/th>\n<th>1\/2<\/th>\n<th>1<\/th>\n<th>2<\/th>\n<th>4<\/th>\n<th>8<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>log_2(x)<\/td>\n<td>-2<\/td>\n<td>-1<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u0414\u043b\u044f y = log_(1\/3)(x) (\u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435 \u043c\u0435\u0436\u0434\u0443 0 \u0438 1, \u043a\u0440\u0438\u0432\u0430\u044f \u0443\u0431\u044b\u0432\u0430\u0435\u0442):<\/p>\n<table>\n<thead>\n<tr>\n<th>x<\/th>\n<th>1\/9<\/th>\n<th>1\/3<\/th>\n<th>1<\/th>\n<th>3<\/th>\n<th>9<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>log_(1\/3)(x)<\/td>\n<td>2<\/td>\n<td>1<\/td>\n<td>0<\/td>\n<td>-1<\/td>\n<td>-2<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u0412\u0438\u0434\u043d\u043e \u0433\u043b\u0430\u0432\u043d\u043e\u0435 \u043e\u0442\u043b\u0438\u0447\u0438\u0435: \u043e\u0431\u0435 \u043a\u0440\u0438\u0432\u044b\u0435 \u043f\u0440\u043e\u0445\u043e\u0434\u044f\u0442 \u0447\u0435\u0440\u0435\u0437 (1; 0), \u043d\u043e \u043f\u0440\u0438 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0438 \u0431\u043e\u043b\u044c\u0448\u0435 1 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044f \u0440\u0430\u0441\u0442\u0443\u0442 \u0432\u043f\u0440\u0430\u0432\u043e, \u0430 \u043f\u0440\u0438 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0438 \u043c\u0435\u043d\u044c\u0448\u0435 1 &#8212; \u0443\u0431\u044b\u0432\u0430\u044e\u0442. \u0427\u0435\u043c \u0431\u043b\u0438\u0436\u0435 x \u043a \u043d\u0443\u043b\u044e, \u0442\u0435\u043c \u043a\u0440\u0443\u0447\u0435 \u0433\u0440\u0430\u0444\u0438\u043a \u043f\u0440\u0438\u0436\u0438\u043c\u0430\u0435\u0442\u0441\u044f \u043a \u0432\u0435\u0440\u0442\u0438\u043a\u0430\u043b\u044c\u043d\u043e\u0439 \u043e\u0441\u0438.<\/p>\n<h2 id=\"s5\">\u041e\u0442\u043a\u0443\u0434\u0430 \u0432 IT \u0431\u0435\u0440\u0435\u0442\u0441\u044f O(log n)<\/h2>\n<p>\u041a\u043b\u044e\u0447\u0435\u0432\u0430\u044f \u0434\u043b\u044f \u043f\u0440\u043e\u0433\u0440\u0430\u043c\u043c\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u044f \u0447\u0435\u0440\u0442\u0430 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0430: \u043e\u043d \u0440\u0430\u0441\u0442\u0435\u0442 \u043e\u0447\u0435\u043d\u044c \u043c\u0435\u0434\u043b\u0435\u043d\u043d\u043e. \u0427\u0442\u043e\u0431\u044b log_2(n) \u0432\u044b\u0440\u043e\u0441 \u043d\u0430 \u0435\u0434\u0438\u043d\u0438\u0446\u0443, \u0430\u0440\u0433\u0443\u043c\u0435\u043d\u0442 n \u043d\u0443\u0436\u043d\u043e \u0443\u0434\u0432\u043e\u0438\u0442\u044c. \u041e\u0431\u0440\u0430\u0442\u043d\u0430\u044f \u0441\u0442\u043e\u0440\u043e\u043d\u0430 \u0442\u043e\u0433\u043e \u0436\u0435 \u0444\u0430\u043a\u0442\u0430 &#8212; \u0435\u0441\u043b\u0438 \u043d\u0430 \u043a\u0430\u0436\u0434\u043e\u043c \u0448\u0430\u0433\u0435 \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c \u0432\u0434\u0432\u043e\u0435 \u0441\u043e\u043a\u0440\u0430\u0449\u0430\u0435\u0442 \u043e\u0431\u044a\u0435\u043c \u0434\u0430\u043d\u043d\u044b\u0445, \u0447\u0438\u0441\u043b\u043e \u0448\u0430\u0433\u043e\u0432 \u0440\u0430\u0432\u043d\u043e \u043f\u0440\u0438\u043c\u0435\u0440\u043d\u043e log_2(n). \u041e\u0442\u0441\u044e\u0434\u0430 \u0438 \u0431\u0435\u0440\u0435\u0442\u0441\u044f \u0441\u043b\u043e\u0436\u043d\u043e\u0441\u0442\u044c O(log n).<\/p>\n<p>\u041a\u043b\u0430\u0441\u0441\u0438\u0447\u0435\u0441\u043a\u0438\u0439 \u043f\u0440\u0438\u043c\u0435\u0440 &#8212; \u0434\u0432\u043e\u0438\u0447\u043d\u044b\u0439 \u043f\u043e\u0438\u0441\u043a \u0432 \u043e\u0442\u0441\u043e\u0440\u0442\u0438\u0440\u043e\u0432\u0430\u043d\u043d\u043e\u043c \u043c\u0430\u0441\u0441\u0438\u0432\u0435: \u0441\u0440\u0430\u0432\u043d\u0438\u0432\u0430\u0435\u043c \u0438\u0441\u043a\u043e\u043c\u043e\u0435 \u0441\u043e \u0441\u0440\u0435\u0434\u043d\u0438\u043c \u044d\u043b\u0435\u043c\u0435\u043d\u0442\u043e\u043c \u0438 \u043a\u0430\u0436\u0434\u044b\u0439 \u0440\u0430\u0437 \u043e\u0442\u0431\u0440\u0430\u0441\u044b\u0432\u0430\u0435\u043c \u043f\u043e\u043b\u043e\u0432\u0438\u043d\u0443. \u041c\u0430\u043a\u0441\u0438\u043c\u0430\u043b\u044c\u043d\u043e\u0435 \u0447\u0438\u0441\u043b\u043e \u0441\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u0439 \u0434\u043b\u044f \u043c\u0430\u0441\u0441\u0438\u0432\u0430 \u0438\u0437 n \u044d\u043b\u0435\u043c\u0435\u043d\u0442\u043e\u0432 \u0440\u0430\u0432\u043d\u043e floor(log_2(n)) + 1.<\/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: #C678DD\">def<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #61AFEF; font-weight: bold\">binary_search_steps<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">data<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E06C75\">target<\/span><span style=\"color: #ABB2BF\">):<\/span>\n    <span style=\"color: #E06C75\">lo<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E06C75\">hi<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E06C75\">steps<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #D19A66\">0<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E5C07B\">len<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">data<\/span><span style=\"color: #ABB2BF\">)<\/span> <span style=\"color: #56B6C2\">-<\/span> <span style=\"color: #D19A66\">1<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">0<\/span>\n    <span style=\"color: #C678DD\">while<\/span> <span style=\"color: #E06C75\">lo<\/span> <span style=\"color: #56B6C2\">&lt;=<\/span> <span style=\"color: #E06C75\">hi<\/span><span style=\"color: #ABB2BF\">:<\/span>\n        <span style=\"color: #E06C75\">steps<\/span> <span style=\"color: #56B6C2\">+=<\/span> <span style=\"color: #D19A66\">1<\/span>\n        <span style=\"color: #E06C75\">mid<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">lo<\/span> <span style=\"color: #56B6C2\">+<\/span> <span style=\"color: #E06C75\">hi<\/span><span style=\"color: #ABB2BF\">)<\/span> <span style=\"color: #56B6C2\">\/\/<\/span> <span style=\"color: #D19A66\">2<\/span>\n        <span style=\"color: #C678DD\">if<\/span> <span style=\"color: #E06C75\">data<\/span><span style=\"color: #ABB2BF\">[<\/span><span style=\"color: #E06C75\">mid<\/span><span style=\"color: #ABB2BF\">]<\/span> <span style=\"color: #56B6C2\">==<\/span> <span style=\"color: #E06C75\">target<\/span><span style=\"color: #ABB2BF\">:<\/span>\n            <span style=\"color: #C678DD\">return<\/span> <span style=\"color: #E06C75\">steps<\/span>\n        <span style=\"color: #C678DD\">if<\/span> <span style=\"color: #E06C75\">data<\/span><span style=\"color: #ABB2BF\">[<\/span><span style=\"color: #E06C75\">mid<\/span><span style=\"color: #ABB2BF\">]<\/span> <span style=\"color: #56B6C2\">&lt;<\/span> <span style=\"color: #E06C75\">target<\/span><span style=\"color: #ABB2BF\">:<\/span>\n            <span style=\"color: #E06C75\">lo<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #E06C75\">mid<\/span> <span style=\"color: #56B6C2\">+<\/span> <span style=\"color: #D19A66\">1<\/span>\n        <span style=\"color: #C678DD\">else<\/span><span style=\"color: #ABB2BF\">:<\/span>\n            <span style=\"color: #E06C75\">hi<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #E06C75\">mid<\/span> <span style=\"color: #56B6C2\">-<\/span> <span style=\"color: #D19A66\">1<\/span>\n    <span style=\"color: #C678DD\">return<\/span> <span style=\"color: #E06C75\">steps<\/span>  <span style=\"color: #7F848E\"># \u043d\u0435 \u043d\u0430\u0439\u0434\u0435\u043d\u043e - \u0432\u0435\u0440\u043d\u0435\u043c \u0447\u0438\u0441\u043b\u043e \u043f\u0440\u043e\u0432\u0435\u0440\u0435\u043d\u043d\u044b\u0445 \u0448\u0430\u0433\u043e\u0432<\/span>\n\n<span style=\"color: #E06C75\">n<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #D19A66\">1_000_000<\/span>\n<span style=\"color: #E06C75\">data<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #E5C07B\">list<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E5C07B\">range<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">n<\/span><span style=\"color: #ABB2BF\">))<\/span>\n<span style=\"color: #E06C75\">worst<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #E5C07B\">max<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">binary_search_steps<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">data<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E06C75\">t<\/span><span style=\"color: #ABB2BF\">)<\/span> <span style=\"color: #C678DD\">for<\/span> <span style=\"color: #E06C75\">t<\/span> <span style=\"color: #56B6C2\">in<\/span> <span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #D19A66\">0<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E06C75\">n<\/span> <span style=\"color: #56B6C2\">-<\/span> <span style=\"color: #D19A66\">1<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E06C75\">n<\/span> <span style=\"color: #56B6C2\">\/\/<\/span> <span style=\"color: #D19A66\">2<\/span><span style=\"color: #ABB2BF\">))<\/span>\n<span style=\"color: #E5C07B\">print<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">worst<\/span><span style=\"color: #ABB2BF\">)<\/span>                          <span style=\"color: #7F848E\"># 20<\/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\">floor<\/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: #E06C75\">n<\/span><span style=\"color: #ABB2BF\">))<\/span> <span style=\"color: #56B6C2\">+<\/span> <span style=\"color: #D19A66\">1<\/span><span style=\"color: #ABB2BF\">)<\/span>   <span style=\"color: #7F848E\"># 20<\/span><\/code><\/pre>\n<p>\u0424\u0430\u043a\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0439 \u0432\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\">20<\/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\">20<\/code>. \u0421\u043c\u044b\u0441\u043b: \u0447\u0442\u043e\u0431\u044b \u043d\u0430\u0439\u0442\u0438 \u043b\u044e\u0431\u043e\u0439 \u0438\u0437 \u043c\u0438\u043b\u043b\u0438\u043e\u043d\u0430 \u044d\u043b\u0435\u043c\u0435\u043d\u0442\u043e\u0432, \u0434\u0432\u043e\u0438\u0447\u043d\u043e\u043c\u0443 \u043f\u043e\u0438\u0441\u043a\u0443 \u0445\u0432\u0430\u0442\u0430\u0435\u0442 20 \u0441\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u0439. \u041b\u0438\u043d\u0435\u0439\u043d\u044b\u0439 \u043f\u0435\u0440\u0435\u0431\u043e\u0440 \u0432 \u0445\u0443\u0434\u0448\u0435\u043c \u0441\u043b\u0443\u0447\u0430\u0435 \u0441\u0434\u0435\u043b\u0430\u043b \u0431\u044b \u043c\u0438\u043b\u043b\u0438\u043e\u043d. \u0422\u043e\u0442 \u0436\u0435 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0438\u0447\u0435\u0441\u043a\u0438\u0439 \u0440\u043e\u0441\u0442 \u0441\u0442\u043e\u0438\u0442 \u0437\u0430 \u0432\u044b\u0441\u043e\u0442\u043e\u0439 \u0441\u0431\u0430\u043b\u0430\u043d\u0441\u0438\u0440\u043e\u0432\u0430\u043d\u043d\u043e\u0433\u043e \u0434\u0432\u043e\u0438\u0447\u043d\u043e\u0433\u043e \u0434\u0435\u0440\u0435\u0432\u0430 \u043f\u043e\u0438\u0441\u043a\u0430, \u0433\u043b\u0443\u0431\u0438\u043d\u043e\u0439 \u043f\u0438\u0440\u0430\u043c\u0438\u0434\u044b (heap) \u0438 \u0440\u0430\u0437\u0431\u0438\u0435\u043d\u0438\u0435\u043c \u0437\u0430\u0434\u0430\u0447 \u0432 \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c\u0430\u0445 \u00ab\u0440\u0430\u0437\u0434\u0435\u043b\u044f\u0439 \u0438 \u0432\u043b\u0430\u0441\u0442\u0432\u0443\u0439\u00bb.<\/p>\n<p>\u041e\u0433\u043e\u0432\u043e\u0440\u043a\u0430: O(log n) \u043e\u043f\u0438\u0441\u044b\u0432\u0430\u0435\u0442 \u043f\u043e\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u043f\u0440\u0438 \u0431\u043e\u043b\u044c\u0448\u0438\u0445 n \u0438 \u0441\u043a\u0440\u044b\u0432\u0430\u0435\u0442 \u043f\u043e\u0441\u0442\u043e\u044f\u043d\u043d\u044b\u0435 \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u0438 \u0438 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0430 &#8212; \u043d\u0430 \u043f\u0440\u0430\u043a\u0442\u0438\u043a\u0435 \u0432\u0430\u0436\u043d\u044b \u0435\u0449\u0435 \u0438 \u043a\u043e\u043d\u0441\u0442\u0430\u043d\u0442\u044b, \u043a\u0435\u0448 \u0438 \u0440\u0435\u0430\u043b\u044c\u043d\u044b\u0435 \u0434\u0430\u043d\u043d\u044b\u0435. \u0410\u0441\u0438\u043c\u043f\u0442\u043e\u0442\u0438\u043a\u0430 \u043f\u043e\u043a\u0430\u0437\u044b\u0432\u0430\u0435\u0442 \u0442\u0435\u043d\u0434\u0435\u043d\u0446\u0438\u044e \u0440\u043e\u0441\u0442\u0430, \u0430 \u043d\u0435 \u0442\u043e\u0447\u043d\u043e\u0435 \u0432\u0440\u0435\u043c\u044f.<\/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, \u0432 \u043a\u043e\u0442\u043e\u0440\u0443\u044e \u043d\u0430\u0434\u043e \u0432\u043e\u0437\u0432\u0435\u0441\u0442\u0438 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435 a, \u0447\u0442\u043e\u0431\u044b \u043f\u043e\u043b\u0443\u0447\u0438\u0442\u044c \u0430\u0440\u0433\u0443\u043c\u0435\u043d\u0442 b; \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0435 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u043e \u0442\u043e\u043b\u044c\u043a\u043e \u043f\u0440\u0438 b &gt; 0, a &gt; 0, a \u2260 1.<\/li>\n<li>\u0422\u0440\u0438 \u0440\u0430\u0431\u043e\u0447\u0438\u0445 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u044f: 2 (\u0434\u0432\u043e\u0438\u0447\u043d\u044b\u0439, \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c\u044b), e \u2248 2,71828 (\u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u044b\u0439, \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u044b\u0439 \u0440\u043e\u0441\u0442), 10 (\u0434\u0435\u0441\u044f\u0442\u0438\u0447\u043d\u044b\u0439, \u0438\u043d\u0436\u0435\u043d\u0435\u0440\u043d\u044b\u0435 \u0448\u043a\u0430\u043b\u044b); \u043f\u0435\u0440\u0435\u0445\u043e\u0434 \u043c\u0435\u0436\u0434\u0443 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u044f\u043c\u0438 &#8212; \u0434\u0435\u043b\u0435\u043d\u0438\u0435\u043c \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u043e\u0432.<\/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 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u044f) \u0434\u0435\u0439\u0441\u0442\u0432\u0443\u044e\u0442 \u043f\u0440\u0438 \u043f\u043e\u043b\u043e\u0436\u0438\u0442\u0435\u043b\u044c\u043d\u044b\u0445 \u0430\u0440\u0433\u0443\u043c\u0435\u043d\u0442\u0430\u0445 \u0438 \u043f\u0440\u0435\u0432\u0440\u0430\u0449\u0430\u044e\u0442 \u0443\u043c\u043d\u043e\u0436\u0435\u043d\u0438\u0435 \u0432 \u0441\u043b\u043e\u0436\u0435\u043d\u0438\u0435.<\/li>\n<li>\u0413\u0440\u0430\u0444\u0438\u043a y = log_a(x): \u043e\u0431\u043b\u0430\u0441\u0442\u044c \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f (0; +\u221e), \u043f\u0440\u043e\u0445\u043e\u0434\u0438\u0442 \u0447\u0435\u0440\u0435\u0437 (1; 0), \u0432\u0435\u0440\u0442\u0438\u043a\u0430\u043b\u044c\u043d\u0430\u044f \u0430\u0441\u0438\u043c\u043f\u0442\u043e\u0442\u0430 x = 0; \u043f\u0440\u0438 a &gt; 1 \u0432\u043e\u0437\u0440\u0430\u0441\u0442\u0430\u0435\u0442, \u043f\u0440\u0438 0 &lt; a &lt; 1 \u0443\u0431\u044b\u0432\u0430\u0435\u0442.<\/li>\n<li>\u0412 IT \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \u0437\u0430\u0434\u0430\u0435\u0442 \u043c\u0435\u0434\u043b\u0435\u043d\u043d\u044b\u0439 \u0440\u043e\u0441\u0442: \u0435\u0441\u043b\u0438 \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c \u0432\u0434\u0432\u043e\u0435 \u0441\u043e\u043a\u0440\u0430\u0449\u0430\u0435\u0442 \u0434\u0430\u043d\u043d\u044b\u0435 \u0437\u0430 \u0448\u0430\u0433, \u0435\u0433\u043e \u0441\u043b\u043e\u0436\u043d\u043e\u0441\u0442\u044c &#8212; O(log n) (\u0434\u0432\u043e\u0438\u0447\u043d\u044b\u0439 \u043f\u043e\u0438\u0441\u043a, \u0434\u0435\u0440\u0435\u0432\u044c\u044f \u043f\u043e\u0438\u0441\u043a\u0430, \u043a\u0443\u0447\u0438).<\/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 \u0442\u043e\u043b\u044c\u043a\u043e \u0448\u043a\u043e\u043b\u044c\u043d\u0430\u044f \u0442\u0435\u043c\u0430: \u0431\u0435\u0437 \u043d\u0438\u0445 \u043d\u0435 \u043e\u0446\u0435\u043d\u0438\u0442\u044c \u0441\u043b\u043e\u0436\u043d\u043e\u0441\u0442\u044c \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c\u0430, \u043d\u0435 \u0440\u0430\u0437\u043e\u0431\u0440\u0430\u0442\u044c \u0441\u0442\u0440\u0443\u043a\u0442\u0443\u0440\u044b \u0434\u0430\u043d\u043d\u044b\u0445 \u0438 \u043d\u0435 \u043f\u043e\u043d\u044f\u0442\u044c, \u043f\u043e\u0447\u0435\u043c\u0443 \u043e\u0434\u043d\u0438 \u0440\u0435\u0448\u0435\u043d\u0438\u044f \u043c\u0430\u0441\u0448\u0442\u0430\u0431\u0438\u0440\u0443\u044e\u0442\u0441\u044f, \u0430 \u0434\u0440\u0443\u0433\u0438\u0435 \u043d\u0435\u0442. \u0422\u043e\u0442 \u0436\u0435 \u0430\u043f\u043f\u0430\u0440\u0430\u0442 \u043d\u0443\u0436\u0435\u043d \u0432 \u0430\u043d\u0430\u043b\u0438\u0442\u0438\u043a\u0435 \u0434\u0430\u043d\u043d\u044b\u0445 \u0438 \u043c\u0430\u0448\u0438\u043d\u043d\u043e\u043c \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0438 &#8212; \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0438\u0447\u0435\u0441\u043a\u0438\u0435 \u0448\u043a\u0430\u043b\u044b, \u043a\u0440\u043e\u0441\u0441-\u044d\u043d\u0442\u0440\u043e\u043f\u0438\u044f, \u043b\u043e\u0433\u0438\u0441\u0442\u0438\u0447\u0435\u0441\u043a\u0430\u044f \u0440\u0435\u0433\u0440\u0435\u0441\u0441\u0438\u044f \u043e\u043f\u0438\u0440\u0430\u044e\u0442\u0441\u044f \u043d\u0430 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0430.<\/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>\u0415\u0441\u043b\u0438 \u0445\u043e\u0447\u0435\u0442\u0441\u044f \u0441\u043e\u0431\u0440\u0430\u0442\u044c \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0443 \u043f\u043e\u0434 \u0437\u0430\u0434\u0430\u0447\u0438 \u0440\u0430\u0437\u0440\u0430\u0431\u043e\u0442\u043a\u0438 \u0438 \u0430\u043d\u0430\u043b\u0438\u0437\u0430 \u0434\u0430\u043d\u043d\u044b\u0445 \u0432 \u0441\u0438\u0441\u0442\u0435\u043c\u0443, \u0430 \u043d\u0435 \u0443\u0447\u0438\u0442\u044c \u0444\u043e\u0440\u043c\u0443\u043b\u044b \u0432\u0440\u0430\u0437\u043d\u043e\u0431\u043e\u0439, \u043f\u043e\u0441\u043c\u043e\u0442\u0440\u0438\u0442\u0435 \u043a\u0443\u0440\u0441 <a href=\"https:\/\/otus.ru\/lessons\/prikladnaya-matematika\/?int_article=logarifmy-opisanie-i-graficheskoe-izobrazhenie&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<\/a>: \u0442\u0430\u043c \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u044b, \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0438 \u043e\u0446\u0435\u043d\u043a\u0430 \u0441\u043b\u043e\u0436\u043d\u043e\u0441\u0442\u0438 \u0440\u0430\u0437\u0431\u0438\u0440\u0430\u044e\u0442\u0441\u044f \u0432 \u0441\u0432\u044f\u0437\u043a\u0435 \u0441 \u043f\u0440\u0430\u043a\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u043c\u0438 \u0437\u0430\u0434\u0430\u0447\u0430\u043c\u0438. \u041f\u043e\u043d\u044f\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\u043c\u043e\u0433\u0430\u044e\u0442 \u0431\u0435\u0441\u043f\u043b\u0430\u0442\u043d\u044b\u0435 <a href=\"https:\/\/otus.ru\/events\/?int_article=logarifmy-opisanie-i-graficheskoe-izobrazhenie&amp;int_place=article&amp;int_variant=v1\">\u043e\u0442\u043a\u0440\u044b\u0442\u044b\u0435 \u0443\u0440\u043e\u043a\u0438<\/a> &#8212; \u044d\u0442\u043e \u0436\u0438\u0432\u043e\u0439 \u0440\u0430\u0437\u0431\u043e\u0440 \u0442\u0435\u043c\u044b \u0441 \u043f\u0440\u0435\u043f\u043e\u0434\u0430\u0432\u0430\u0442\u0435\u043b\u0435\u043c.<\/p>\n<\/div>\n<p>\u0421\u043c\u0435\u0436\u043d\u044b\u0435 \u0442\u0435\u043c\u044b: <a href=\"https:\/\/otus.ru\/journal\/logarifmy-i-ih-osnovnye-svojstva-chto-nuzhno-znat-specialistu\/\">\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<\/a>.<\/p>\n<h2 id=\"s8\">FAQ<\/h2>\n<p><strong>\u041f\u043e\u0447\u0435\u043c\u0443 \u043d\u0435\u043b\u044c\u0437\u044f \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c\u0438\u0440\u043e\u0432\u0430\u0442\u044c \u043d\u043e\u043b\u044c \u0438 \u043e\u0442\u0440\u0438\u0446\u0430\u0442\u0435\u043b\u044c\u043d\u044b\u0435 \u0447\u0438\u0441\u043b\u0430?<\/strong> \u041b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \u0438\u0449\u0435\u0442 \u043f\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c \u0441\u0442\u0435\u043f\u0435\u043d\u0438, \u0430 \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 \u0441\u0442\u0435\u043f\u0435\u043d\u0438 \u0434\u0430\u0435\u0442 \u0442\u043e\u043b\u044c\u043a\u043e \u043f\u043e\u043b\u043e\u0436\u0438\u0442\u0435\u043b\u044c\u043d\u043e\u0435 \u0447\u0438\u0441\u043b\u043e. \u041f\u043e\u044d\u0442\u043e\u043c\u0443 \u0434\u043b\u044f \u043d\u0443\u043b\u044f \u0438 \u043e\u0442\u0440\u0438\u0446\u0430\u0442\u0435\u043b\u044c\u043d\u044b\u0445 \u0430\u0440\u0433\u0443\u043c\u0435\u043d\u0442\u043e\u0432 \u0442\u0430\u043a\u043e\u0433\u043e \u043f\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044f \u043d\u0435 \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u0435\u0442, \u0438 \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0435 \u043d\u0435 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u043e.<\/p>\n<p><strong>\u0427\u0442\u043e \u0437\u043d\u0430\u0447\u0438\u0442 \u0437\u0430\u043f\u0438\u0441\u044c log \u0431\u0435\u0437 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u044f?<\/strong> \u0412 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0435 \u043f\u043e \u043a\u043e\u043d\u0442\u0435\u043a\u0441\u0442\u0443: \u0447\u0430\u0441\u0442\u043e lg (\u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435 10) \u0438\u043b\u0438 ln (\u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435 e). \u0412 \u0442\u0435\u043a\u0441\u0442\u0430\u0445 \u043f\u043e \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c\u0430\u043c log n \u043f\u043e\u0447\u0442\u0438 \u0432\u0441\u0435\u0433\u0434\u0430 \u043e\u0437\u043d\u0430\u0447\u0430\u0435\u0442 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435 2 &#8212; \u043f\u0440\u0438 \u043e\u0446\u0435\u043d\u043a\u0435 O(&#8230;) \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0435 \u043c\u0435\u043d\u044f\u0435\u0442 \u043b\u0438\u0448\u044c \u043f\u043e\u0441\u0442\u043e\u044f\u043d\u043d\u044b\u0439 \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044c \u0438 \u0435\u0433\u043e \u043e\u043f\u0443\u0441\u043a\u0430\u044e\u0442.<\/p>\n<p><strong>\u041a\u0430\u043a \u043f\u043e\u0441\u0447\u0438\u0442\u0430\u0442\u044c \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \u043f\u043e \u0440\u0435\u0434\u043a\u043e\u043c\u0443 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u044e \u043d\u0430 \u043a\u0430\u043b\u044c\u043a\u0443\u043b\u044f\u0442\u043e\u0440\u0435 \u0438\u043b\u0438 \u0432 \u043a\u043e\u0434\u0435, \u0435\u0441\u043b\u0438 \u0435\u0441\u0442\u044c \u0442\u043e\u043b\u044c\u043a\u043e ln \u0438 lg?<\/strong> \u0427\u0435\u0440\u0435\u0437 \u0444\u043e\u0440\u043c\u0443\u043b\u0443 \u043f\u0435\u0440\u0435\u0445\u043e\u0434\u0430: log_a(x) = ln(x) \/ ln(a) (\u0438\u043b\u0438 lg(x) \/ lg(a)). \u041d\u0430\u043f\u0440\u0438\u043c\u0435\u0440, log_2(10) = ln(10) \/ ln(2).<\/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 c, \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: \u0435\u0441\u043b\u0438 a^c = b, \u0442\u043e log_a(b) = c. \u041f\u0440\u043e\u0449\u0435 \u0433\u043e\u0432\u043e\u0440\u044f, \u043b\u043e\u0433\u0430\u0440\u0438\u0444\u043c \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 \u0430\u0440\u0433\u0443\u043c\u0435\u043d\u0442\u00bb. \u041d\u0438\u0436\u0435 \u0440\u0430\u0437\u0431\u0435\u0440\u0435\u043c \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0441 \u0443\u0441\u043b\u043e\u0432\u0438\u044f\u043c\u0438, \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u0438 \u0444\u043e\u0440\u043c\u0443\u043b\u044b, \u0442\u0440\u0438 \u0445\u043e\u0434\u043e\u0432\u044b\u0445 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u044f (2, e, 10), 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