# How do you rewrite log_x(3/10) as a ratio of common logs and natural logs?

Nov 24, 2017

$\log \frac{\frac{3}{10}}{\log} x = - \frac{0.5228}{\log} x = - \frac{1.204}{\ln} x$

#### Explanation:

We use the identity ${\log}_{a} b = \log \frac{b}{\log} a$ (common logs - base $10$) or $\ln \frac{b}{\ln} a$ (natural logs).

Hence converting ${\log}_{x} \left(\frac{3}{10}\right)$ in common logs, we have

$\log \frac{\frac{3}{10}}{\log} x$

= $\frac{\log 3 - \log 10}{\log} x$

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

= $\frac{0.4771 - 1}{o} g x = - \frac{0.5228}{\log} x$

and converting ${\log}_{x} \left(\frac{3}{10}\right)$ to natural logs, we have

$\ln \frac{\frac{3}{10}}{\ln} x$

= $\frac{\ln 0.3}{\ln} x$

= $- \frac{1.204}{\ln} x$