Solution assignment 10 Exponential equations

Solve:

$\displaystyle\frac{1}{\sqrt{2\pi}}e^{-x^2}=\frac{1}{8}$

We rewrite the equation:

$e^{-x^2}=\displaystyle\frac{\sqrt{2\pi}}{8}$

$-x^2\ln(e)=\ln(\displaystyle\frac{\sqrt{2\pi}}{8})$

$x^2=-\ln(\displaystyle\frac{\sqrt{2\pi}}{8})=\ln(\displaystyle\frac{8}{\sqrt{2\pi}})$

and thus the solution is:

$x=\sqrt{\ln(\displaystyle\frac{8}{\sqrt{2\pi}})}$

or:

$x=-\sqrt{\ln(\displaystyle\frac{8}{\sqrt{2\pi}})}$