# How do you solve (1/5)^x=10?

Sep 10, 2015

$x = \log \frac{10}{\log 1 - \log 5}$

#### Explanation:

By taking logarithms in both sides we get

$\log {\left(\frac{1}{5}\right)}^{x} = \log 10 \implies x \left(\log 1 - \log 5\right) = \log 10 \implies x = \log \frac{10}{\log 1 - \log 5}$

Sep 10, 2015

Use properties of exponents and logs to reformulate and solve, finding:

$x = - \frac{1}{\log} \left(5\right)$

#### Explanation:

${\left(\frac{1}{5}\right)}^{x} = {5}^{- x}$

So $\log \left({\left(\frac{1}{5}\right)}^{x}\right) = \log \left({5}^{- x}\right) = - x \cdot \log \left(5\right)$

$\log \left(10\right) = 1$

So taking logs of both sides, the original equation becomes:

$- x \cdot \log \left(5\right) = 1$

Divide both sides by $- \log \left(5\right)$ to get:

$x = - \frac{1}{\log} \left(5\right)$