Change of base for a base-4 logarithm

Statement

Compute $\log_4 8$ by changing the base to base $2$. Show the steps without using a calculator.

Solution

Using the change-of-base formula, $\log_4 8=\dfrac{\log_2 8}{\log_2 4}$. Since $8=2^3$ and $4=2^2$ we have $\log_2 8=3$ and $\log_2 4=2$, hence $\log_4 8=\frac{3}{2}.$

$\boxed{\log_4 8=\dfrac{3}{2}}$