Please solve this equation.2^x=x^2

Question

Dear My friends

Please solve this exponential equation for me

 

x^2=2^x

thaks

Answer 1

 

 

     X^{2  =} 2^X

 

     By trying and error, we see that only X = 2 is the solution

 

                   

       2 ²  = 2²

 

       If we take  :   2 log ₂X = Xlog₂ 2 = X    X=2 is the solution.

                                                  

Answer 2

y = x^2 is a parabola, and y = 2^x is an exponential function. If you sketch the two functions, you can tell that there are two intersection points.

x = 2 is an easy one to get by guessing and error.

The other one can be obtained by numerical method such as Newton's method.

y = 2^x - x^2

y' = 2^x (ln2) - 2x

x(n+1) = x(n) - y(n)/y'(n)

x(0) = -1

x(1) = -.7869

x(2) = -.7668

x(3) = -.7667

...

It converges to -.766664696