x^2 + 9 = (x + 9)^2
Expanding the right side:
x^2 + 9 = x^2 + 18x + 81
Rearranging the equation:
0 = 18x + 72
18x = -72
x = -4

(5x - 8)^2 = (5x - 2)^2
Expanding both sides:
25x^2 - 80x + 64 = 25x^2 - 20x + 4
Rearranging the equation:
60x = 60
x = 1

log^5(x^2 + 5x) = log^5(x^2 + 9)
Since the bases are the same, we can equate the arguments:
x^2 + 5x = x^2 + 9
5x = 9
x = 9/5

(x - 8)^2 = (x + 9)^2
Expanding both sides:
x^2 - 16x + 64 = x^2 + 18x + 81
Rearranging the equation:
-34x = 17
x = -1/2

Therefore, the solutions to the equations are x = -4, x = 1, x = 9/5, and x = -1/2.