# How do you solve  ln(x-4)-ln(3x-10)=ln (1/x)?

Sep 7, 2016

$x = 5$

#### Explanation:

Use the rule ${\log}_{a} \left(n\right) - {\log}_{a} \left(m\right) = {\log}_{a} \left(\frac{n}{m}\right)$.

$\implies \ln \left(\frac{x - 4}{3 x - 10}\right) = \ln \left(\frac{1}{x}\right)$

Now use the rule $\ln a = \ln b \to a = b$.

$\frac{x - 4}{3 x - 10} = \frac{1}{x}$

${x}^{2} - 4 x = 3 x - 10$

${x}^{2} - 7 x + 10 = 0$

$\left(x - 5\right) \left(x - 2\right) = 0$

$x = 5 \mathmr{and} 2$

However, $x = 2$ is extraneous, since it renders the equation undefined.

Hopefully this helps!