How do you rewrite log_(1/3)x as a ratio of common logs and natural logs?

Nov 26, 2017

I tried this:

Explanation:

We can change into an new base $c$ base as:

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

so in your case we get:

• common log:

${\log}_{\frac{1}{3}} x = {\log}_{10} \frac{x}{\log} _ \left(10\right) \left(\frac{1}{3}\right)$

• natural log:

${\log}_{\frac{1}{3}} x = {\log}_{e} \frac{x}{\log} _ e \left(\frac{1}{3}\right)$