# How do you rewrite log_(1/5)x as a ratio of common logs and natural logs?

Feb 23, 2017

${\log}_{b} a = {\log}_{c} \frac{a}{\log} _ c b$
${\log}_{\frac{1}{5}} x = {\log}_{10} \frac{x}{\log} _ 10 \left(\frac{1}{5}\right)$
${\log}_{\frac{1}{5}} x = \ln \frac{x}{\ln} \left(\frac{1}{5}\right)$

#### Explanation:

You would apply the formula:

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

Then, for example,

${\log}_{\frac{1}{5}} x = {\log}_{10} \frac{x}{\log} _ 10 \left(\frac{1}{5}\right)$

and

${\log}_{\frac{1}{5}} x = \ln \frac{x}{\ln} \left(\frac{1}{5}\right)$