Question

O logaritmo de um número na base 8 é igual a -2. Calcule o logaritmo desse número na base 16. Ajudem pf ?

Answer 1

Descobrindo x primeiro:

$log_{8}(x)=-2\\8^{-2}=x\\(2^{3})^{-2}=x\\x=2^{-6}$

Achando log de x na base 16:

$log_{16}(2^{-6})=y\\16^{y}=2^{-6}\\(2^{4})^{y}=2^{-6}\\2^{4y}=-6\\4y=-6\\y=-6/4\\\\\boxed{\boxed{y=-\dfrac{3}{2}}}$
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Sem descobrir x (mais trabalhoso):

$log_{8}(x)=-2$

Mudando a base pra 16:

$\dfrac{log_{16}(x)}{log_{16}(8)}=-2\\\\\\\dfrac{log_{16}(x)}{log_{(2^{4})}(2^{3})}=-2\\\\\\\dfrac{log_{16}(x)}{3\cdot(\frac{1}{4})\cdot log_{2}(2)}=-2\\\\\\\dfrac{log_{16(x)}}{(\frac{3}{4})}=-2\\\\\\log_{16}(x)=(-2)\cdot\dfrac{3}{4}\\\\\\\boxed{\boxed{log_{16}(x)=-\dfrac{3}{2}}}$