solution for: x to the power logx equal to 100x?

Answer 1

x^(logx) = 100x

Take log of both sides:

log(x^(logx)) = log 100x

Power rule:

logx(logx) = (logx)^2 = log100x

Multiplication rule:

(logx)^2 = log100 + logx

Get everything on one side:

(logx)^2 - logx - log100 = 0

Just to make it look better, log100 = 2, so

(logx)^2 - logx - 2 = 0

Let u = logx. Then:

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

So:

logx = 2
x = 100

or

logx = -1
x = 0.1

Answer 2

Well,

x^(log x) = x = 100x

100x-x = 0
99x = 0
x = 0, invalid solution

There's really no solution because you can't take the log of 0.

It might have a limiting value.

Answer 3

x ^log x=100x
log x * log x=log 100 + log x
(log x)^2-log x -2=0
(log x -2)(log x+1)=0
log x=2
x=100

log x=-1
x=0.1

roots are x=2, 0.1