# How do you solve lnx=5-ln2x ?

Apr 11, 2016

$x = 8.614$, rounded to three decimal places.

#### Explanation:

To solve $\ln x = 5 - \ln 2 x$, for $x$

Rearranging to bring all terms containing $x$ to left hand side of the equation we obtain
$\ln x + \ln 2 x = 5$
Using the property of logarithms that $\log \left(a \times b\right) = \log a + \log b$ we obtain
$\ln \left(x \times 2 x\right) = 5$
or $\ln 2 {x}^{2} = 5$

Using the definition of log functions we obtain

${e}^{5} = 2 {x}^{2}$, solving for $x$
$x = \pm \sqrt{{e}^{5} / 2}$, ignoring the $- v e$ sign as logarithms of a negative number is not defined.

$x = \sqrt{{e}^{5} / 2}$, with a calculator

or $x = 8.614$, rounded to three decimal places.