# How do you rewrite log_x(3/4) as a ratio of common logs and natural logs?

Mar 3, 2017

In common logarithm ${\log}_{x} \left(\frac{3}{4}\right) = \log \frac{\frac{3}{4}}{\log} x$

and in natural logarithm ${\log}_{x} \left(\frac{3}{4}\right) = \ln \frac{\frac{3}{4}}{\ln} x$

#### Explanation:

Let ${\log}_{b} a = x$, then we know that ${b}^{x} = a$

and taking common logs (i.e. to the base $10$) on both sides, we get

$\log {b}^{x} = \log a$ i.e. $x \log b = \log a$ or $x = \log \frac{a}{\log} b$

As $x = {\log}_{b} a$, we can say ${\log}_{b} a = \log \frac{a}{\log} b$

Hence in common logarithm ${\log}_{x} \left(\frac{3}{4}\right) = \log \frac{\frac{3}{4}}{\log} x$

Now as ${b}^{x} = a$, taking natural logarithm we get

$\ln {b}^{x} = \ln a$ i.e. $x \ln b = \ln a$ or $x = \ln \frac{a}{\ln} b$

i.e. ${\log}_{x} \left(\frac{3}{4}\right) = \ln \frac{\frac{3}{4}}{\ln} x$