# Question #04717

Mar 23, 2017

Given: $\ln \left(\frac{x - 5}{x - 1}\right) = \ln \left(6\right)$

Subtract $\ln \left(6\right)$ from both sides:

$\ln \left(\frac{x - 5}{x - 1}\right) - \ln \left(6\right) = 0$

Use the property of logarithms where subtracting is equivalent to division within the argument:

$\ln \left(\frac{x - 5}{6 \left(x - 1\right)}\right) = 0$

Make both sides the exponent of e:

${e}^{\ln \left(\frac{x - 5}{6 \left(x - 1\right)}\right)} = {e}^{0}$

The logarithm and the exponential on the left side cancel and the right side becomes 1:

$\frac{x - 5}{6 \left(x - 1\right)} = 1$

Now, it is easy to solve for x:

$x - 5 = 6 x - 6$

$5 x = 1$

$x = \frac{1}{5}$

Check:

$\ln \left(\frac{\frac{1}{5} - 5}{\frac{1}{5} - 1}\right) = \ln \left(6\right)$

$\ln \left(\frac{1 - 25}{1 - 5}\right) = \ln \left(6\right)$

$\ln \left(\frac{- 24}{- 4}\right) = \ln \left(6\right)$

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

This checks.

$x = \frac{1}{5}$

Mar 24, 2017

$x = \frac{1}{5}$

#### Explanation:

The contents of the two logarithms, which are the only things present on either side of the equation, are equal:

$\frac{x - 5}{x - 1} = 6$

$\implies x - 5 = 6 \left(x - 1\right)$

$\implies x - 5 = 6 x - 6$

$\implies 1 = 5 x$

$\implies x = \frac{1}{5}$