Question

log 2 16 – log 4 32 é igual a:

Answer 1

Pra facilitar, vamos calcular os logaritmos separadamente e, posteriormente, podemos calcular o valor da expressão.

Seja, log[2]16=x  e   log[4]32=y, temos:

$\log_{_2}16~=~x\\\\\\Aplicando~a~\underline{de finicao~de~logaritmo}\\\\\\16~=~2^x\\\\\\Temos~uma~equacao~exponencial\\Vamos~igualar~as~bases~nos~dois~lados~da~equacao\\Reescrevendo~16~como~uma~\underline{potencia~de~base~2}\\\\\\2^4~=~2^x\\\\\\Bases~iguais,~igualamos~os~expoentes\\\\\\2\!\!\!\backslash^4~=~2\!\!\!\backslash^x\\\\\\\boxed{x~=~4}$

$\log_{_2}32~=~y\\\\\\Aplicando~a~\underline{de finicao~de~logaritmo}\\\\\\32~=~4^y\\\\\\Temos~uma~equacao~exponencial\\Vamos~igualar~as~bases~nos~dois~lados~da~equacao\\Reescrevendo~32~e~4~como~\underline{potencias~de~base~2}\\\\\\2^5~=~\left(2^2\right)^y\\\\\\2^5~=~2^{2y}\\\\\\Bases~iguais,~igualamos~os~expoentes\\\\\\2\!\!\!\backslash^5~=~2\!\!\!\backslash^{2y}\\\\\\5~=~2y\\\\\\\boxed{y~=~\dfrac{5}{2}~~ou~~ 2,5}$

Podemos agora achar o valor da expressão:

$\log_{_2}16~-~\log_{_4}32~=\\\\\\=~x~-~y\\\\\\=~4~-~2,5\\\\\\=~\boxed{~1,5~}$

Resposta:  1,5