Question

como resolver log 128 na base 256?

Answer 1

$x=\mathrm{\ell og}_{256\,}128\;\;\Leftrightarrow\;\;256^{x}=128\\ \\ \\ 256^{x}=128\\ \\ \left(2^{8} \right )^{x}=2^{7}\\ \\ 2^{8x}=2^{7}\\ \\ 8x=7\\ \\ x=\dfrac{7}{8}\;\;\Rightarrow\;\;\boxed{\mathrm{\ell og}_{256\,}128=\dfrac{7}{8}}$

Answer 2

Resolverei por outro método, uma vez que o convencional já foi usado

Propriedades:

$log_{b}(a^{n})=n\cdot log_{b}(a)\\\\log_{(b^{n})}(a)=\frac{1}{n}\cdot log_{b}(a)\\\\log_{a}(a)=1$
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$log_{256}(128)=log_{(2^{8})}(2^{7})\\\\log_{256}(128)=7\cdot log_{(2^{8})}(2)\\\\log_{256}(128)=7\cdot\frac{1}{8}\cdot log_{2}(2)\\\\log_{256}(128)=\frac{7}{8}\cdot1\\\\\\\boxed{\boxed{log_{256}(128)=\dfrac{7}{8}}}$