# How do you solve 2^x = 5^(x - 2)?

Apr 10, 2016

I found: $x = \frac{2 \ln \left(5\right)}{\left(\ln \left(5\right) - \ln \left(2\right)\right)} = 3.513$

#### Explanation:

I would take the natural log of both sides:

$\textcolor{red}{\ln} {2}^{x} = \textcolor{red}{\ln} {5}^{x - 2}$

then use the fact that $\log {x}^{m} = m \log x$ and write:
$x \ln \left(2\right) = \left(x - 2\right) \ln \left(5\right)$

rearrange:

$x \ln \left(2\right) - x \ln \left(5\right) = - 2 \ln \left(5\right)$
$x \left[\ln \left(2\right) - \ln \left(5\right)\right] = - 2 \ln \left(5\right)$

and:

$x = \frac{- 2 \ln \left(5\right)}{\left(\ln \left(2\right) - \ln \left(5\right)\right)} = \frac{2 \ln \left(5\right)}{\left(\ln \left(5\right) - \ln \left(2\right)\right)} = 3.513$