# How can I find X? (Exponential Equation)

## ${2}^{3 x + 1} = {3}^{x - 2}$

##### 1 Answer
Apr 13, 2017

$x = - \frac{\ln 2 + 2 \ln 3}{3 \ln 2 - \ln 3}$

#### Explanation:

Given:

${2}^{3 x + 1} = {3}^{x - 2}$

Take logs of both sides to get:

$\left(3 x + 1\right) \ln 2 = \left(x - 2\right) \ln 3$

Multiply out to get:

$\left(3 \ln 2\right) x + \ln 2 = \left(\ln 3\right) x - 2 \ln 3$

Subtract $\ln 2 + \left(\ln 3\right) x$ from both sides to get:

$\left(3 \ln 2 - \ln 3\right) x = - \ln 2 - 2 \ln 3$

Divide both sides by $\left(3 \ln 2 - \ln 3\right)$ to get:

$x = - \frac{\ln 2 + 2 \ln 3}{3 \ln 2 - \ln 3}$