Question

Calcule a soma de S no seguinte caso:
S = log de 0,001 na base 100 + log de 4/9 na base 3/2 - log de raiz cúbica de 10 na base 100

Answer 1

Ae manu,

para calcular a expressão, 

$S=\log_{100}0,001+ \log_{ \tfrac{3}{2}} \dfrac{4}{9}-\log_{100} \sqrt[3]{10}$

temos que lembrar de algumas propriedades de logaritmos..

PROPRIEDADE DECORRENTE DA DEFINIÇÃO, A D1:

$\log_bb=1$

PROPRIEDADE DA POTÊNCIA, A P3:

$\log(b)^n=n\cdot \log(b)$

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$S=\log_{100}0,001+ \log_{ \tfrac{3}{2}} \dfrac{4}{9}-\log_{100} \sqrt[3]{10}\\\\ S=\log_{10^2}\left( \dfrac{1}{1000}\right)+\log_{ \tfrac{3}{2}}\left( \dfrac{2}{3}^2\right)-\log_{10^2}\left(10^{ \tfrac{1}{3}\right)\\\\ S=\log_{10^2} \left( \dfrac{1}{10^3}\right)+\log_{ \tfrac{3}{2}}\left( \dfrac{2}{3}\right)^2-\log_{10^2}(10)^{ \tfrac{1}{3}}$

$S=\log_{10^2}(10)^{-3}+2\cdot\log_{ \tfrac{3}{2}}\left( \dfrac{2}{3} \right)- \dfrac{1}{3}\cdot \log_{10^2}}(10)\\\\ S=(-3)\cdot\log_{10^2}}(10)+2\log_{ \tfrac{3}{2}}\left( \dfrac{2}{3}\right)- \dfrac{1}{3}\log_{10^2}}(10)\\\\ S=2\cdot(-3)\log_{10}(10)-2\log_{ \tfrac{3}{2}}}\left( \dfrac{3}{2}\right)-2\cdot \dfrac{1}{3}\log_{10}(10)\\\\ S=(-6)\cdot1-2\cdot1- \dfrac{2}{3}\cdot1\\\\ \Large\boxed{S=- \dfrac{26}{3} }$


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tenha ótimos
estudos ^^