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.

The equation will have only one solution, namely

**x = 10**

10^10 = 10^10

To prove the uniqueness of this solution, the equation could be solved drawing the graphs of the exponential function, 10^x, and the polynomial function, x^10.

The intercepting point of these 2 graphs is the solution of the equation.

To draw each graph, we'll have to input values to x.