# How do you evaluate log_20 0.125 using the change of base formula?

Apr 15, 2018

${\log}_{20} 0.125 = - 0.6941$

#### Explanation:

According to change of base formula

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

and when we use base $10$, we do not write $10$ in subscript.

Hence ${\log}_{20} 0.125$

= $\log \frac{0.125}{\log} 20$

= $\log \frac{\frac{1}{8}}{\log} \left(2 \cdot 10\right)$

= $\frac{\log 1 - \log 8}{\log 2 + \log 10}$

= $\frac{- \log {2}^{3}}{\log 2 + 1}$

= $- \frac{3 \log 2}{1 + \log 2}$

= $- \frac{3 \cdot 0.3010}{1.3010}$

= $- 0.6941$