Question

Se log² = 0,3 e log³ = 0,5 , calcule:

a) log 1,5
b) log 1/4
c) log 72​

Answer 1

Nesse exercício, vamos reescrever os logaritmandos de tal forma que possamos utilizar os dados fornecidos no enunciado juntamente as propriedades de logaritmos.

a)

$\log1,5~=\\\\\\Reescrevendo~o~logaritmando~na~sua~\underline{forma~fracionaria~reduzida}:\\\\\\=~\log\left(\dfrac{15}{10}\right)\\\\\\=~\log\left(\dfrac{3}{2}\right)\\\\\\Aplicando~a~propriedade~do~\underline{logaritmo~do~quociente}:\\\\\\=~\log3~-~\log2\\\\\\Substituindo~os~valores:\\\\\\=~0,5~-~0,3\\\\\\=~\boxed{~0,2~}$

b)

$\log\left(\dfrac{1}{4}\right)~=\\\\\\Aplicando~a~propriedade~do~\underline{logaritmo~do~quociente}:\\\\\\=~\log1~-~\log4\\\\\\\underline{Fatorando}~o~logaritmando~4:\\\\\\=~\log1~-~\log\,(2\cdot2)\\\\\\Aplicando~a~propriedade~do~\underline{logaritmo~do~produto}:\\\\\\=~log1~-~\left(\log2~+~\log2\right)\\\\\\Substituindo~os~valores:\\\\\\=~0~-~(0,3~+~0,3)\\\\\\=~\boxed{-0,6}$

c)

$\log72~=\\\\\\\underline{Fatorando}~o~logaritmando:\\\\\\=~\log\,(2\cdot2\cdot2\cdot3\cdot 3)\\\\\\Aplicando~a~propriedade~do~\underline{logaritmo~do~produto}:\\\\\\=~\log2~+~\log2~+~\log2~+~\log3~+~\log3\\\\\\Substituindo~os~valores:\\\\\\=~0,3~+~0,3~+~0,3~+~0,5~+~0,5\\\\\\=~\boxed{~1,9~}$

$\Huge{\begin{array}{c}\Delta \tt{\!\!\!\!\!\!\,\,o}\!\!\!\!\!\!\!\!\:\,\perp\end{array}}Qualquer~d\acute{u}vida,~deixe~ um~coment\acute{a}rio$