Question

3. Sabe-se que Log 7 = 0,84 e Log 3 = 0,47. Sendo assim, calcule: a) Log 42/18 b) Log 21 c) Log 63 d) Log 1/70

Answer 1

Neste exercício vamos utilizar a fatoração e propriedades de logaritmos e potências.

a)

$\log\left(\dfrac{42}{18}\right)~=\\\\\\\tt{Fatorando~o~numerador~e~denominador~do~logaritmando:}\\\\\\=~\log\left(\dfrac{2\cdot3\cdot 7}{2\cdot 3\cdot 3}\right)\\\\\\=~\log\left(\dfrac{\backslash\!\!\!2\cdot \backslash\!\!\!3\cdot 7}{\backslash\!\!\!2\cdot \backslash\!\!\!3\cdot 3}\right)\\\\\\=~\log\left(\dfrac{7}{3}\right)\\\\\\Utilizando~a~propriedade~do~logaritmo~do~quociente:\\\\\\=~\log7~-~\log3\\\\\\Substituindo~os~valores~dos~logaritmos:\\\\\\=~0,84~-~0,47$

$=~\boxed{0,37}$

b)

$\log21~=\\\\\\\tt{Fatorando~o~logaritmando:}\\\\\\=~\log\left(3\cdot 7\right)\\\\\\Utilizando~a~propriedade~do~logaritmo~do~produto:\\\\\\=~\log3~+~\log7\\\\\\Substituindo~os~valores~dos~logaritmos:\\\\\\=~0,47~+~0,84\\\\\\=~\boxed{1,31}$

c)

$\log63~=\\\\\\\tt{Fatorando~o~logaritmando:}\\\\\\=~\log\left(3\cdot3\cdot 7\right)\\\\\\Utilizando~a~propriedade~do~logaritmo~do~produto:\\\\\\=~\log3~+~\log3~+~\log7\\\\\\Substituindo~os~valores~dos~logaritmos:\\\\\\=~0,47~+~0,47~+~0,84\\\\\\=~\boxed{1,78}$

d)

$\log21~=\\\\\\\tt{Fatorando~o~denominador~do~logaritmando:}\\\\\\=~\log\left(\dfrac{1}{7\cdot 10}\right)\\\\\\Utilizando~a~propriedade~do~logaritmo~do ~quociente\\\\\\=~\log1~-~\big(\log(7\cdot10)\big)\\\\\\Utilizando~a~propriedade~do~logaritmo~do ~produto\\\\\\=~\log1~-~(\log7~+~\log10)\\\\\\Substituindo~os~valores~dos~logaritmos:\\(lembre~que~\log1=0~e~\log10=1)\\\\\\=~0~-~(0,84~+~1)\\\\\\=~\boxed{-1,84}$

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