Question #c3090

Apr 5, 2016

$x = 3$

Explanation:

Use the identity

$\ln \left(a\right) - \ln \left(b\right) \equiv \ln \left(\frac{a}{b}\right)$

Therefore,

$\ln \left(4 x\right) - \ln \left(x - 1\right) = \ln \left(\frac{4 x}{x - 1}\right)$

$= \ln \left(6\right)$

Since $\ln$ is an inverse function, we take the exponential with base $e$ on both sides to "undo" the $\ln$.

$\ln \left(\frac{4 x}{x - 1}\right) = \ln \left(6\right)$

${e}^{\ln \left(\frac{4 x}{x - 1}\right)} = {e}^{\ln \left(6\right)}$

$\frac{4 x}{x - 1} = 6$

Simplify the equation

$\frac{4 x}{x - 1} = 6$

$\frac{2 x}{x - 1} = 3$

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

$= 3 x - 3$

$0 = x - 3$

$x = 3$