How to convert the base $10$ logarithm to base $2$?
The question didn't provide the value of the logarithm.
The question is: "Given a logarithm in base 10, how can you use the change of base property to convert it to a logarithm in base 2?"
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The rule is $\log_a(x)=\frac{\log_b(x)}{\log_b(a)}$. Can you identify $a$ and $b$ in your case? – Scene Jul 04 '23 at 11:12
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Recall that by definition
$$x=\log_{10}y \iff 10^x=y$$
now let
$$z=\log_{2}10 \iff 2^z=10$$
then
$$(2^z)^x=2^{xz}=y$$
therefore
$$xz=\log_2 y \implies x=\frac{\log_2 y}{z} \implies \log_{10}y=\frac{\log_2 y}{\log_2{10}}$$
and more in general (for positive $a,b,c \neq 1$)
$$\log_{a}b=\frac{\log_c b}{\log_c{a}}$$
user
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You can use the property of base transformation here. Like this:-
$$ \log_{10}x = \frac{\log_2 x}{\log_2 10} $$
Pyking
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2Welcome to stackexchange. That said, please don't answer questions where the OP shows no work of their own. – Ethan Bolker Jul 04 '23 at 11:16