# What is the value of x for which ln(2x-5)-lnx=1/4?

Dec 4, 2017

$x = \frac{5}{2 - {e}^{\frac{1}{4}}} \approx 6.9835$

#### Explanation:

$\ln \left(2 x - 5\right) - \ln x = \frac{1}{4}$

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

$\ln \left(\frac{2 x - 5}{x}\right) = \frac{1}{4}$

Taking antilogarithm:

${e}^{\ln} \left(\frac{2 x - 5}{x}\right) = {e}^{\frac{1}{4}}$

$\frac{2 x - 5}{x} = {e}^{\frac{1}{4}}$

$\left(2 x - 5\right) = x {e}^{\frac{1}{4}}$

$2 x - x {e}^{\frac{1}{4}} = 5$

$x \left(2 - {e}^{\frac{1}{4}}\right) = 5$

$x = \frac{5}{2 - {e}^{\frac{1}{4}}} \approx 6.9835$