Question

calcule os logaritmos urgente é p amanhã !!!

a)log $\frac{1}{3} ^{243}$

b)log $\frac{5}{3} ^{ \frac{27}{125} }$

c)log $\frac{2}{3} ^{ \frac{729}{64} }$

Answer 1

Oi Maysa ;D

use as seguintes propriedades de log:

$logb^k=k\cdot logb\\\\ log_kk=1\\\\log_{\tfrac{1}{b}}=-log_b$

__________________

$log_{ \tfrac{1}{3}}243=-log_33^5\\\\ log_{ \tfrac{1}{3}}243=5\cdot(-log_33)\\\\ log_{ \tfrac{1}{3}}243=5\cdot(-1)\\\\ \Large\boxed{\boxed{\boxed{log_{ \tfrac{1}{3}}243=-5}}}$

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$log_{ \tfrac{5}{3}} \dfrac{27}{125}=log_{ \tfrac{5}{3}} \dfrac{27}{125}\\\\ log_{ \tfrac{5}{3}} \dfrac{27}{125}=log_{ \tfrac{5}{3}}\dfrac{3^3}{5^3}\\\\ log_{ \tfrac{5}{3}} \dfrac{27}{125}=log_{ \tfrac{5}{3}}\left( \dfrac{3}{5}\right)^3\\\\ log_{ \tfrac{5}{3}} \dfrac{27}{125}=log_{ \tfrac{5}{3}}\left( \dfrac{5}{3}\right)^{-3}\\\\ log_{ \tfrac{5}{3}} \dfrac{27}{125}=(-3)\cdot log_{ \tfrac{5}{3}} \dfrac{5}{3}\\\\ log_{ \tfrac{5}{3}} \dfrac{27}{125}=(-3)\cdot1$

$\Large\boxed{\boxed{\boxed{log_{ \tfrac{5}{3}} \dfrac{27}{125}=-3}}}.\\.$

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$log_{ \tfrac{2}{3}} \dfrac{729}{64}= log_{ \tfrac{2}{3} }\dfrac{3^6}{2^6}\\\\ log_{ \tfrac{2}{3} } \dfrac{729}{64}=log_{ \tfrac{2}{3} } \dfrac{2}{3}^{-6}\\\\ log_{ \tfrac{2}{3}} \dfrac{729}{64} =(-6)\cdot log_{ \tfrac{2}{3} } \dfrac{2}{3}\\\\\\ \Large\boxed{\boxed{\boxed{log_{ \tfrac{2}{3} } \dfrac{729}{64}=-6}}}.\\.$

Tenha ótimos estudos ^^