# How do you evaluate log_(1/3) (1/81)?

Dec 16, 2016

$4$

#### Explanation:

$\frac{1}{81}$ can be written as ${\left(81\right)}^{-} 1$:

$= {\log}_{\frac{1}{3}} \left({81}^{-} 1\right)$

Use the change of base rule ${\log}_{a} n = \log \frac{n}{\log} a$:

$= \log {81}^{-} \frac{1}{\log} \left(\frac{1}{3}\right)$

$= \log {\left({3}^{4}\right)}^{-} \frac{1}{\log} \left({3}^{-} 1\right)$

$= \log \frac{{3}^{-} 4}{\log} {3}^{-} 1$

Use the exponent rule that $\log {a}^{n} = n \log a$:

$= \frac{- 4 \log 3}{- 1 \log 3}$

$= 4$

Hopefully this helps!