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

Mar 4, 2016

$x = 3.513$

#### Explanation:

To solve 2^x=5^(x−2), take logs of both the sides, this becomes

$x \log 2 = \left(x - 2\right) \log 5 = x \log 5 - 2 \log 5$ and transposing terms this becomes

$x \log 5 - x \log 2 = 2 \log 5$ or

$x \left(\log 5 - \log 2\right) = 2 \log 5$

$x = \frac{2 \log 5}{\log 5 - \log 2} = \frac{2 \times 0.699}{0.699 - 0.301}$ or

$x = \frac{1.398}{0.398} = 3.513$