# How do you solve 12^(x-4)=3^(x-2)?

Jan 7, 2017

$x = 5.585$

#### Explanation:

As ${12}^{x - 4} = {3}^{x - 2}$, taking log on both sides, we get

$\left(x - 4\right) \log 12 = \left(x - 2\right) \log 3$

or $x \log 12 - 4 \log 12 = x \log 3 - 2 \log 3$

or $x \log 12 - x \log 3 = 4 \log 12 - 2 \log 3$

or $x \left(\log 12 - \log 3\right) = \log \left(\frac{{12}^{4}}{{3}^{2}}\right)$

or $x \log 4 = \log \left({4}^{2} \cdot {12}^{2}\right) = 2 \log 48$

or $x = \frac{2 \log 48}{\log 4} = \frac{2 \times 1.6812}{0.6021} = 5.585$