# How do you solve 2^(x+1) = 3^x?

Feb 26, 2016

Convert to logarithmic form.

#### Explanation:

$\log {2}^{x + 1} = \log {3}^{x}$

Simplify using the rule $\log {a}^{n} = n \log a$

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

Distribute on the left side. Don't forget you cannot multiply directly a log with a non-log (e.g $2 \times \log 3 \ne \log 6$but is equal to $2 \log 3$)

$x \log 2 + \log 2 = x \log 3$

Put the x's to one side of the equation.

$\log 2 = x \log 3 - x \log 2$

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

Simplify the right side of the equation further by using the rule $\log m - \log n = \log \left(\frac{m}{n}\right)$

$\log 2 = x \left(\log \left(\frac{3}{2}\right)\right)$

$\log \frac{2}{\log} \left(\frac{3}{2}\right) = x$

You will want to ask your teacher if he/she wants the answer in exact form or rounded off. Just make sure to check.

Practice exercises:

1. Solve for x. Leave answers in exact form.

a) ${2}^{3 x} = {5}^{x + 1}$

b). ${5}^{x - 3} = {3}^{2 x + 1}$

Challenge problem

Find the value of x in ${2}^{4 x - 6} = 5 \times {3}^{x + 7}$