# How do you calculate  log_5(4) ?

Apr 8, 2016

${\log}_{5} \left(4\right) = 0.8614$

#### Explanation:

Let ${\log}_{b} a = x$, then ${b}^{x} = a$.

If $a = {10}^{n}$ and $b = {10}^{m}$, then $n = \log a$ and $m = \log b$ and

${b}^{x} = a$ becomes ${\left({10}^{m}\right)}^{x} = {10}^{n}$ or ${10}^{m x} = {10}^{n}$ i.e.

$m x = n$

Hence $x = \frac{n}{m} = \log \frac{a}{\log} b$

Thus ${\log}_{5} \left(4\right) = \log \frac{4}{\log} 5 = \frac{0.6021}{0.6990} = 0.8614$