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

Dec 1, 2015

$x = - \frac{4 \ln \left(3\right)}{3 \ln \left(4\right) - \ln \left(3\right)}$

#### Explanation:

For this problem, we will be using the property of logarithms that

$\ln \left({a}^{x}\right) = x \ln \left(a\right)$

${4}^{3 x} = {3}^{x - 4}$

$\implies \ln \left({4}^{3 x}\right) = \ln \left({3}^{x - 4}\right)$

#=> 3xln(4) = (x-4)ln(3)

$\implies 3 x \ln \left(4\right) - x \ln \left(3\right) = - 4 \ln \left(3\right)$

$\implies x \left(3 \ln \left(4\right) - \ln \left(3\right)\right) = - 4 \ln \left(3\right)$

$\implies x = - \frac{4 \ln \left(3\right)}{3 \ln \left(4\right) - \ln \left(3\right)}$