What is x for 5^-6 = 5^(x^2 - 5x) ?

This is an exponential
equation.

We'll write the equation again using symmetric
property:

5^(x^2-5x) =
5^-6

We'll use the one to one property, because the bases
are matching:

x^2 - 5x =
-6

We'll add 6 both sides:

x^2
- 5x + 6 = 0

We'll apply the quadratic
formula:

x1 = [5+sqrt(25 -
24)]/2

x1 =
(5+1)/2

x1 =
3

x2 =
(5-1)/2

x2 =
2

Verifying:

For
x1 = 3:

5^(3^2-5*3) =
5^-6

5^(9-15) = 5^-6

5^-6 =
5^-6

For x2 = 2:

5^(2^2-5*2) =
5^-6

5^(4-10) = 5^-6

5^-6 =
5^-6

We'll not reject any of found
solutions.