# How do you solve e ^(lnx-3lnx) = (5/2)?

Apr 17, 2016

$x = \sqrt{\frac{2}{5}}$

#### Explanation:

Note that $x$ cannot be negative.

Taking log of both sides to base $e$ we get

$\ln x - 3 \ln x = \ln \left(\frac{5}{2}\right)$ now simplifying LHS

$\ln x - 3 \ln x = \ln x - \ln {x}^{3} = \ln \left(\frac{x}{x} ^ 3\right) = \ln \left(\frac{1}{x} ^ 2\right) = \ln \left(\frac{5}{2}\right)$

Hence $\frac{1}{x} ^ 2 = \frac{5}{2}$ or ${x}^{2} = \frac{2}{5}$

Hence $x = \pm \sqrt{\frac{2}{5}}$ but x cannot be negative.

Hence $x = \sqrt{\frac{2}{5}}$.