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

Oct 29, 2016

How you do it is explained below.

#### Explanation:

Take the natural logarithm of both sides:

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

Use the identity $\ln \left({a}^{b}\right) = \ln \left(a\right) b$

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

Distribute ln(5):

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

Subtract ln(5)x from both sides:

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

Multiply both sides by -1:

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

Divide both sides by $\left(\ln \left(5\right) - \ln \left(2\right)\right)$:

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

$x \approx 3.5$