# How do you solve  log x = x^2 - 2?

May 27, 2016

$- i \sqrt{W \frac{- \frac{2}{e} ^ 4}{2}} \mathmr{and} - i \sqrt{W \frac{- 1 , - \frac{2}{e} ^ 4}{2}}$

#### Explanation:

This is where it gets messy

There is not elementary way to solve this problem

Fortunately I do have a solution

But first lets see why there is no scope to solve this using logarithmic properties

$\log x = {x}^{2} - 2$

I am going to raise 10 to the power of each side

${10}^{\log} x = {10}^{{x}^{2} - 2}$

$x = {10}^{{x}^{2} - 2}$

Now if you want to try solving this you would probably end up going in cirles

# So here is where the real solution start

Have you heard about the Lambert W function or product log

Here is a brief explanation

HERE

So lets Begin

log

log⁡(x)=x^2−2

Another thing In your case I am assuming logarithm to be Natural logarithm

$\ln x = {x}^{2} - 2$

$x = {e}^{{x}^{2} - 2}$

$x \cdot {e}^{2 - {x}^{2}} = 1$

Now apply the the Lambert W function on both the sides

Since this is very big calculation I used wolfram alpha

So the 2 roots are

$- i \sqrt{W \frac{- \frac{2}{e} ^ 4}{2}} \mathmr{and} - i \sqrt{W \frac{- 1 , - \frac{2}{e} ^ 4}{2}}$