{"id":94133,"date":"2017-01-11T09:06:10","date_gmt":"2017-01-11T11:06:10","guid":{"rendered":"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/?p=94133"},"modified":"2026-05-26T16:28:06","modified_gmt":"2026-05-26T18:28:06","slug":"logaritmo","status":"publish","type":"post","link":"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/matematica\/logaritmo","title":{"rendered":"Logaritmo"},"content":{"rendered":"<p><strong>Logaritmo<\/strong>\u00a0\u00e9 uma ferramenta matem\u00e1tica utilizada para tornar opera\u00e7\u00f5es mais simples.<\/p>\n<p><b>Logaritmo<\/b> \u00e9 a opera\u00e7\u00e3o matem\u00e1tica inversa da exponencia\u00e7\u00e3o, que calcula o expoente necess\u00e1rio para elevar uma base a um determinado n\u00famero, representado por <strong>log\u00a0<sub>a<\/sub> b = x<\/strong> se e somente se <strong>a<sup>x<\/sup> = b<\/strong>. Ele \u00e9 essencial em c\u00e1lculos exponenciais, finan\u00e7as, biologia e qu\u00edmica, facilitando contas complexas.<\/p>\n<p><b>Logaritmo &#8211;\u00a0<\/b><strong>Defini\u00e7\u00e3o<\/strong><\/p>\n<p><strong>O\u00a0<\/strong><strong>logaritmo<\/strong><strong>\u00a0de um n\u00famero b na base a \u00e9 o expoente x ao qual a deve ser elevado para produzir b.<\/strong><\/p>\n<p><strong>Logaritmo \u00e9 a opera\u00e7\u00e3o matem\u00e1tica que busca encontrar o expoente a que uma base fixa deve ser elevada para resultar em um determinado n\u00famero (chamado logaritmando).<\/strong><\/p>\n<p>Sejam a e b n\u00fameros reais positivos e b \u2260\u00a01. Chama-se\u00a0logaritmo de a na base b o expoente x tal que b<sup>x<\/sup>\u00a0= a.<\/p>\n<p class=\"imagem\"><img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone wp-image-122297 size-full\" title=\"Logaritmo\" src=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/logaritmo.png\" alt=\"Logaritmo\" width=\"340\" height=\"150\" srcset=\"https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/logaritmo.png 340w, https:\/\/lab.fabricadelandings.com.br\/portalsaofrancisconovo\/wp-content\/uploads\/2017\/01\/logaritmo-300x132.png 300w\" sizes=\"(max-width: 340px) 100vw, 340px\" \/><\/p>\n<p>log\u00a0<sub>2<\/sub>\u00a016 = x , \u00e9 o expoente x tal que 2<sup>x<\/sup>\u00a0= 16<\/p>\n<p><b>2<sup>x<\/sup>\u00a0=\u00a02<sup>4 \u00a0<\/sup>:<\/b>. x = 4<\/p>\n<p>Desse modo,\u00a0log\u00a0<sub>2<\/sub>\u00a016 =\u00a04.<\/p>\n<p>O inventor do logaritmo foi\u00a0John Napier, que durante o s\u00e9culo XVII trabalhou muito para que se encontrasse os logaritmos. Juntamente com Napier,\u00a0Henry Briggs tamb\u00e9m se dedicou aos estudos sobre o assunto e junto criaram a t\u00e1bua de logaritmos.<\/p>\n<p>Na \u00e9poca que estudaram essa importante ferramenta do c\u00e1lculo n\u00e3o haviam calculadoras, e ent\u00e3o essa t\u00e1bua criada por eles s\u00e3o exatamente os logaritmos que utilizamos hoje em dia.<\/p>\n<p><strong>Consequ\u00eancias da Defini\u00e7\u00e3o<\/strong><\/p>\n<p><b>Como consequ\u00eancias da defini\u00e7\u00e3o de logaritmo, temos as seguintes propriedades:<\/b><\/p>\n<p><strong>1)<\/strong>\u00a0O logaritmo da base, em qualquer base,\u00a0\u00e9 igual a 1.<\/p>\n<p><strong>log\u00a0<sub>a<\/sub>\u00a0a\u00a0=\u00a01<\/strong><\/p>\n<p><strong>2)<\/strong>\u00a0O logaritmo da unidade, em qualquer base, \u00e9 igual a 0.<\/p>\n<p><strong>log\u00a0<sub>a<\/sub>\u00a01 =\u00a00<\/strong><\/p>\n<p><strong>3)\u00a0<\/strong>A pot\u00eancia de base a e expoente\u00a0log\u00a0<sub>a<\/sub>\u00a0b \u00e9 igual a b.<\/p>\n<p><strong>a\u00a0<sup>log\u00a0<sub>a<\/sub>\u00a0b<\/sup>\u00a0= b<\/strong><\/p>\n<p><strong>4)<\/strong>\u00a0Dois logaritmos em uma mesma base s\u00e3o iguais, se, e somente se, os logaritmos s\u00e3o iguais.<\/p>\n<p align=\"left\"><strong>\u00a0log\u00a0<sub>a<\/sub>\u00a0b =\u00a0\u00a0log\u00a0<sub>a<\/sub>\u00a0c\u00a0\u21d4 b = c<\/strong><\/p>\n<p><b>Outras Defini\u00e7\u00f5es<\/b><\/p>\n<p><strong>Logaritmo Natural ou Neperiano:<\/strong><\/p>\n<p><strong>log\u00a0<em><sub>e<\/sub><\/em>\u00a0a \u00a0<\/strong>ou \u00a0<strong>ln a<\/strong><\/p>\n<p><strong>Logaritmo Decimal:<\/strong>\u00a0<b>\u00c9 aquele cujo a base \u00e9 10, e \u00e9 representado por<\/b>:<\/p>\n<p><strong>log a<\/strong><\/p>\n<p><strong>Cologaritmo:\u00a0<\/strong>\u00c9 aquele oposto ao log ou o log do inverso do logaritmando.<\/p>\n<p><strong>colog\u00a0<sub>b<\/sub>\u00a0a = \u2013 log\u00a0<sub>b<\/sub>\u00a0a \u00a0 \u00a0 \u00a0 \u00a0\u00a0colog\u00a0<sub>b<\/sub>\u00a0a =\u00a0log\u00a0<sub>b<\/sub>\u00a01\/a<\/strong><\/p>\n<p><strong>Antilogaritmo:\u00a0<\/strong>O antilogaritmo de x na base b, o n\u00famero a, ou seja, o logaritmando.<\/p>\n<p><strong>log\u00a0<sub>b<\/sub>\u00a0a = x\u00a0\u21d4 a = antilog\u00a0<sub>b\u00a0<\/sub>x<\/strong><\/p>\n<p><b>Propriedades dos Logaritmos<\/b><\/p>\n<p><strong>1)<\/strong>\u00a0log\u00a0<sub>b<\/sub>\u00a0b = 1<\/p>\n<p><strong>2)<\/strong>\u00a0log\u00a0<sub>b<\/sub>\u00a01 =\u00a00<\/p>\n<p><strong>3)<\/strong>\u00a0log\u00a0<sub>b<\/sub>\u00a0a<sup>y<\/sup>\u00a0=\u00a0y log\u00a0<sub>b<\/sub>\u00a0a<\/p>\n<p><strong>4)<\/strong>\u00a0log\u00a0<sub>b<\/sub>\u00a0b<sup>x<\/sup>\u00a0=\u00a0x<\/p>\n<p><strong>5)<\/strong>\u00a0b\u00a0<sup>log\u00a0<sub>b<\/sub>\u00a0a<\/sup>\u00a0= a<\/p>\n<p><strong>6)<\/strong>\u00a0log\u00a0<sub>b<\/sub>\u00a0ac = log\u00a0<sub>b<\/sub>\u00a0a +\u00a0log\u00a0<sub>b<\/sub>\u00a0c<\/p>\n<p><strong>7)<\/strong>\u00a0log\u00a0<sub>b<\/sub>\u00a0a\/c =\u00a0log\u00a0<sub>b<\/sub>\u00a0a \u2013\u00a0log\u00a0<sub>b<\/sub>\u00a0c<\/p>\n<p><strong>8)<\/strong>\u00a0<b>Propriedade de Mudan\u00e7a de Base:<\/b><\/p>\n<p>log\u00a0<sub>b<\/sub>\u00a0a = log\u00a0<sub>k<\/sub>\u00a0a \/\u00a0log\u00a0<sub>k<\/sub>\u00a0b<\/p>\n<p>para qualquer k e\u00a0|R *<sub>+<\/sub>, k\u00a0\u2260 1.<\/p>\n<p><b>Exemplos e Exerc\u00edcios<\/b><\/p>\n<p><strong>1)<\/strong>\u00a0Calcule pela defini\u00e7\u00e3o log\u00a0<sub>4<\/sub>\u00a016.<\/p>\n<p><b>Resolu\u00e7\u00e3o:<\/b><\/p>\n<p>log\u00a0<sub>4<\/sub>\u00a016 = x<br \/>\n<b>4<sup>x<\/sup>\u00a0= 16 \u2192 4<sup>x<\/sup>\u00a0=\u00a04<sup>2\u00a0<\/sup>\u2192 x = 2 \u2192 :<\/b>.\u00a0log\u00a0<sub>4<\/sub>\u00a016 = 2.<\/p>\n<p><strong>2)<\/strong>\u00a0Calcule pela defini\u00e7\u00e3o log\u00a0<sub>0,25<\/sub>\u00a032<\/p>\n<p><b>Resolu\u00e7\u00e3o:<\/b><\/p>\n<p>log\u00a0<sub>0,25<\/sub>\u00a032 = x<br \/>\n0,25<sup>x<\/sup>\u00a0= 32<br \/>\n(\u00bc)<sup>x<\/sup>\u00a0= 32<br \/>\n(1\/2\u00b2)<sup>x<\/sup>\u00a0= 2<sup>5<br \/>\n<\/sup>(2<sup>-2<\/sup>)<sup>x<\/sup>\u00a0= 2<sup>5<br \/>\n<\/sup>-2x = 5<br \/>\n<b>x = -5\/2 \u00a0 \u00a0\u2192\u00a0<\/b>:.\u00a0 log\u00a0<sub>0,25<\/sub>\u00a032 = -5\/2.<\/p>\n<p><strong>3)<\/strong>\u00a0Calcule pela defini\u00e7\u00e3o log\u00a0<sub>3<\/sub>\u00a01\/9<\/p>\n<p><b>Resolu\u00e7\u00e3o:<\/b><\/p>\n<p>log\u00a0<sub>3<\/sub>\u00a01\/9 = x<br \/>\n3<sup>x<\/sup>\u00a0= 1\/9<br \/>\n3<sup>x<\/sup>\u00a0= 1\/3\u00b2<br \/>\n3<sup>x<\/sup>\u00a0=\u00a03<sup>-2<br \/>\n<\/sup><b>x = -2 \u00a0\u2192\u00a0<\/b>:.\u00a0log\u00a0<sub>3<\/sub>\u00a01\/9 = -2.<\/p>\n<p><strong>4)<\/strong>\u00a0Calcule log\u00a0<sub>3<\/sub>\u00a05 transformado para a base 2.<\/p>\n<p><b>Resolu\u00e7\u00e3o:<\/b><\/p>\n<p>log\u00a0<sub>3<\/sub>\u00a05 = log\u00a0<sub>2<\/sub>\u00a05 \/\u00a0log\u00a0<sub>2<\/sub>\u00a03<\/p>\n<p><strong>Fonte:<\/strong>\u00a0Portal S\u00e3o Francisco<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Voc\u00ea conhece Logaritmo e suas propriedades?<\/p>\n","protected":false},"author":3,"featured_media":132251,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[414],"tags":[],"class_list":["post-94133","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-matematica"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.5 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Logaritmo, Opera\u00e7\u00e3o, Matem\u00e1tica, N\u00famero Logaritmo<\/title>\n<meta name=\"description\" content=\"Logaritmo, Opera\u00e7\u00e3o, Matem\u00e1tica, N\u00famero, Base, Ferramenta, C\u00e1lculo, Propriedades, Exemplos, Defini\u00e7\u00e3o, Logaritmo\" \/>\n<meta name=\"robots\" content=\"noindex, follow, max-snippet:-1, max-image-preview:large, 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