How to convert a logarithm from base 10 to base 2?

Question

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?"

Answer 1

You can use the property of base transformation here. Like this:-

$$ \log_{10}x = \frac{\log_2 x}{\log_2 10} $$

Answer 2

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 $a,b,c \neq 1$)

$$\log_{a}b=\frac{\log_c b}{\log_c{a}}$$