# How do you evaluate log_(1/3) (1/27)?

Oct 1, 2016

$3$

#### Explanation:

${\log}_{\frac{1}{3}} \left(\frac{1}{27}\right)$
$= {\log}_{\frac{1}{3}} {\left(\frac{1}{3}\right)}^{3}$
$= 3 {\log}_{\frac{1}{3}} \left(\frac{1}{3}\right)$
$= 3$

Alternatively, you can do it this way :

Recall that ${\log}_{a} b = c , \implies {a}^{c} = b$

let ${\log}_{\frac{1}{3}} \left(\frac{1}{27}\right) = x$
$\implies {\left(\frac{1}{3}\right)}^{x} = \frac{1}{27} = {\left(\frac{1}{3}\right)}^{3}$
Hence, $x = 3$