2^x=x^2
2 to the power X equals x to the power 2
I have found a solution i.e 2 instantly but I found many answers so I operated by log which confused me

I'm not sure how you found "many answers" - I believe there are 3.

For positive values of x you can do this:



If you plot you'll see that f(x) equals at only two points, those being x = 2 and x=4. To prove that there are the only two solutions - if you find the derivative of this function you'll see that it equals zero only at x = e, so there can be at most two points where it equals . Hence there are only two solutions for positive values of x.

For negative values of x we can let y = -x, and then a similar approach yields



which has a solution at around y = 0.7667, so x = -0.7667 is the third solution. As a check - below is a plot of and and you can see the three points where the functions cross.