# What is log_4(2sqrt(2)) ?

Mar 26, 2016

${\log}_{4} \left(2 \sqrt{2}\right) = \frac{3}{4}$

#### Explanation:

The change of base formula tells us that if $a , b , c > 0$ and $a , c \ne 1$ then:

${\log}_{a} b = \frac{{\log}_{c} b}{{\log}_{c} a}$

Hence:

${\log}_{4} \left(2 \sqrt{2}\right) = {\log}_{2} \frac{2 \sqrt{2}}{\log} _ 2 \left(4\right) = {\log}_{2} \frac{{2}^{\frac{3}{2}}}{\log} _ 2 \left({2}^{2}\right) = \frac{\frac{3}{2}}{2} = \frac{3}{4}$