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3ª  questão) Desenvolva a expressão utilizando as propriedades operatórias dos logaritmos:​

Answer 1

Desenvolvendo a expressão logarítmica usando as propriedades, obtemos como resultado  $~~\sf\dfrac{~3log\,x~}{2}-\dfrac{~5log\,z~}{4}~$ .

⠀

Dado a expressão logarítmica

                                                  $\Large\quad\quad\quad\begin{array}{l}\sf log\,\sqrt{\dfrac{x^3\sqrt{z}}{z^3}}\\\\\end{array}$

, vamos desenvolver usando as propriedades operatórias dos logaritmos, vide em anexo:

$\begin{array}{l}\sf log\,\sqrt{\dfrac{x^3\sqrt{z}}{z^3}}=log\:\bigg[\dfrac{x^3\sqrt{z}}{z^3}\bigg]^{1/2}\\\\\sf log\,\sqrt{\dfrac{x^3\sqrt{z}}{z^3}}=\dfrac{~1~}{2}\cdot\bigg[log\:\dfrac{x^3\sqrt{z}}{z^3}\bigg]\\\\\sf log\,\sqrt{\dfrac{x^3\sqrt{z}}{z^3}}=\dfrac{~1~}{2}\cdot\big[log\,x^3\sqrt{z}-log\,z^3\big]\\\\\sf log\,\sqrt{\dfrac{x^3\sqrt{z}}{z^3}}=\dfrac{~1~}{2}\cdot\big[log\,x^3+log\,\sqrt{z}-log\,z^3\big]\end{array}$

$\begin{array}{l}\sf log\,\sqrt{\dfrac{x^3\sqrt{z}}{z^3}}=\dfrac{~1~}{2}\cdot\big[log\,x^3+log\,z^{\,1/2}-log\,z^3\big]\\\\\sf log\,\sqrt{\dfrac{x^3\sqrt{z}}{z^3}}=\dfrac{~1~}{2}\cdot\big[3\cdot log\,x+\dfrac{~1~}{2}\cdot log\,z-3\cdot log\,z\big]\\\\\sf log\,\sqrt{\dfrac{x^3\sqrt{z}}{z^3}}= \dfrac{~1~}{2}\cdot\bigg[3log\,x+\dfrac{-3log\,z\cdot2+log\,z~}{2}\bigg]\\\\\sf log\,\sqrt{\dfrac{x^3\sqrt{z}}{z^3}}=\dfrac{~1~}{2}\cdot\bigg[3log\,x+\dfrac{-6log\,z+log\,z~}{2}\bigg]\end{array}$

$\begin{array}{l}\sf log\,\sqrt{\dfrac{x^3\sqrt{z}}{z^3}}=\dfrac{~1~}{2}\cdot\bigg[3log\,x-\dfrac{~5log\,z~}{2}\bigg]\\\\\!\boldsymbol{\boxed{\sf log\,\sqrt{\dfrac{x^3\sqrt{z}}{z^3}}=\dfrac{~3log\,x~}{2}-\dfrac{~5log\,z~}{4}}}\\\\\end{array}$

 

Este é o valor simplificado da expressão inicial.  

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Answer 2

log √[(x³√z)/z³]

log [(x³√z)/z³]^(1/2)

(1/2) * [ log  x³ + log z^(1/2-3) ]  

(1/2) * [ log  x³ + log z^(-5/2) ]  

(1/2) * [ 3 log  x -5/2 log z]