Solve this equation:

Question

x^log(x)=100x

Answer 1

Given: x^log(x)=100xl

 

I will take log of each side: (log x)(log x) = log 100 + log x; Note log 100 = 2

 

let z = log x.       z² - z - 2 = 0.        (z-2)(z+1) = 0.        z = -1 or z = 2.

 

z=2 means log x = 2 so x = 10² = 100.

z=-1 means log x = -1 so x = 10^-1 = 1/10

 

Two answers!

 

Answer 2

Take log of both sides:  log(x^logx) = log(100x)

 

                                  (logx)(logx) = log100 + logx

 

 (logx)² - logx -2 = 0

 

Let u = logx.  Then, u² - u - 2 = 0

 

                             (u-2)(u+1) = 0

 

                             u = 2 or u = -1

 

Since u = logx, we have logx = 2 or logx = -1.

 

So, x = 10² or x = 10^-1

 

x = 100 or 1/10.