# How do you evaluate log_.25 (-4)?

Aug 12, 2016

The principal value of ${\log}_{0.25} \left(- 4\right)$ is $- 1 - \frac{\pi}{\ln \left(4\right)} i$

#### Explanation:

Use the change of base formula to find:

${\log}_{0.25} \left(- 4\right) = \ln \frac{- 4}{\ln} \left(0.25\right) = \ln \frac{- 4}{\ln \left(\frac{1}{4}\right)} = \ln \frac{- 4}{- \ln \left(4\right)} = \frac{\ln \left(4\right) + \pi i}{- \ln \left(4\right)}$

$= - 1 - \frac{\pi}{\ln \left(4\right)} i$

Note this is the principal value of the logarithm.

There are other solutions of ${0.25}^{x} = - 4$, namely:

$x = - 1 - \frac{\pi \left(1 + 2 k\right)}{\ln \left(4\right)} i$

for any integer $k$.