# How do you evaluate log (1/100)?

It is

$\log \left(\frac{1}{100}\right) = \log \left(\frac{1}{10} ^ 2\right) = \log 1 - \log {10}^{2} = - 2 \log 10 = - 2$

Note that ${\log}_{10} 10 = 1$

Mar 11, 2016

$\log \left(\frac{1}{100}\right) = - 2$

#### Explanation:

First, lets assume that the base of the logarithm is $10$, sometimes written ${\log}_{10}$. Next, we'll simplify by using the knowledge that

$\log \left({x}^{a}\right) = a \cdot \log \left(x\right)$

We can convert the $\frac{1}{100}$ in the expression to a power of $10$:

$\log \left(\frac{1}{100}\right) = \log \left({100}^{- 1}\right) = \log \left({\left({10}^{2}\right)}^{- 1}\right) = \log \left({10}^{-} 2\right)$

Which we can rewrite as

$- 2 \cdot \log \left(10\right) = - 2$

since ${\log}_{10} \left(10\right) = 1$