# How do you solve 2^(x+3) = 3^(x-4)?

Sep 24, 2016

$x = 16.4695$

#### Explanation:

As the bases are different, we cannot just compare them.

The variables are in the exponents, so logs are called for.

Log both sides:

$\log {2}^{x + 3} = \log {3}^{x - 4} \text{ } \leftarrow$ use the log power law

$\left(x + 3\right) \log 2 = \left(x - 4\right) \log 3 \text{ } \leftarrow$ move the log terms to one side

$\frac{x + 3}{x - 4} = \frac{\log 3}{\log 2} = 1.58496$

$\frac{x + 3}{x - 4} = 1.58496 \text{ } \leftarrow$ cross-mulitply

$x + 3 = 1.58496 \left(x - 4\right)$

$x + 3 = 1.58496 x - 6.33985 \text{ } \leftarrow$ re-arrange the terms

$3 + 6.33985 = 1.58496 x - x$

$9.633985 = 0.58496 x$

$\frac{9.633985}{0.58496} = x$

$x = 16.4695$