# How do you simplify log_5 (1/250)?

Apr 28, 2016

I found: $- {\log}_{5} \left(2\right) - 3$

#### Explanation:

We can write it as:
${\log}_{5} \left(\frac{1}{250}\right) = {\log}_{5} {\left(250\right)}^{-} 1 = - {\log}_{5} \left(250\right) =$
using the fact that: $\log \left(x y\right) = \log x + \log y$
$= - \log \left(2 \cdot 125\right) = - \left[{\log}_{5} \left(2\right) + {\log}_{5} \left(125\right)\right] =$
using the definition of log we get:
$= - {\log}_{5} \left(2\right) - 3$

or alternatively we can use the fact that:

$\log \left(\frac{x}{y}\right) = \log \left(x\right) - \log \left(y\right)$
and again:
$\log \left(x y\right) = \log x + \log y$
and write:
${\log}_{5} \left(\frac{1}{250}\right) = {\log}_{5} \left(1\right) - {\log}_{5} \left(250\right) =$
$= {\log}_{5} \left(1\right) - {\log}_{5} \left(2 \cdot 125\right) =$
$= {\log}_{5} \left(1\right) - \left[{\log}_{5} \left(2\right) + {\log}_{5} \left(125\right)\right] =$
using the definition of log again we get:
$= 0 - {\log}_{5} \left(2\right) - 3 =$
$= - {\log}_{5} \left(2\right) - 3$