10^x=x^10 find x?

10^x =
x^10.

Solution:

The function
has an obvious  solution at x= 10 .  But It has other solution than x =10. Let us 
examine.

Let f(x) = 10^x -
x^10.

f(1) = 10^1 -1^10 = 9 >
0

f(2) = 10^2 -2^10 = 100 -1024 =
-924.

Therefore f(x) being a continuous and derivable
function f(x) = 0 has a solution in the interval (1 , 2) as it changes the sign for x=1
and x =2.

We can go for an iteration method  and have a
solution for x.

When x =
1.371289,

10^x =23.51197 and x^10 =23.51202. So
f(x) < 0

when x=
1.371288

10^x = 23.51191 and x^10 = 23.51185 . Here
f(x) > 0.

So clearly there is a solution  for x in
between 1.371288 and 1.371289 where f(x) change the sign.