# How do you solve ln x = 2(ln 1 - ln 11)?

Jan 17, 2016

$x = \frac{1}{121}$

#### Explanation:

Divide both sides by $2$.

$\left(\frac{1}{2}\right) \ln x = \ln 1 - \ln 11$

Simplify the right hand side using the logarithm rule: $\ln a - \ln b = \ln \left(\frac{a}{b}\right)$

$\left(\frac{1}{2}\right) \ln x = \ln \left(\frac{1}{11}\right)$

Simplify the left hand side using the logarithm rule: $a \ln x = \ln \left({x}^{a}\right)$

$\ln \left({x}^{\frac{1}{2}}\right) = \ln \left(\frac{1}{11}\right)$

$\ln \sqrt{x} = \ln \left(\frac{1}{11}\right)$

Thus, since if $\ln a = \ln b$, then $a = b$,

$\sqrt{x} = \frac{1}{11}$

$x = {\left(\frac{1}{11}\right)}^{2}$

$x = \frac{1}{121}$