Question

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

Answer 1

`-i sqrt(W(-2/e^4)/2) and -i sqrt(W(-1,-2/e^4)/2)`

Explanation:

This is where it gets messy

There is not elementary way to solve this problem

Probably your best shot at solving this problem is graphing it.And says a a lot about 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^logx = 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

You can read more about this
HERE

So lets Begin

log

`log⁡(x)=x^2−2`

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

`lnx =x^2 -2`

`x = e^(x^2-2)`

`x*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(-2/e^4)/2) and -i sqrt(W(-1,-2/e^4)/2)`