Take the Log

Some exponential equations have no same-base trick. In ${2}^{x} = 10$, the 10 is not a power of 2: ${2}^{3} = 8$ and ${2}^{4} = 16$, so x sits somewhere between 3 and 4, and no whole number works.

A log names that number. It asks for a power, so $\log_{2}\left(2^{x}\right)$ is simply x. Taking log base 2 of both sides turns ${2}^{x} = 10$ into $x = \log_{2}\left(10\right)$. The base of the log is always the base of the power.

$\log_{2}\left(10\right)$ is the exact answer. A calculator gives about 3.32, which is a rounded copy of the same number. Keep the exact form, just as you keep $\sqrt{2}$ rather than 1.41.