(5/2)^x=(4/25)^2

We start with the equation:

  (5/2)^x = (4/25)^2

Step 1. Rewrite (4/25) in terms of squares:
  4 = 2^2 and 25 = 5^2, so
  (4/25) = (2^2 / 5^2) = (2/5)^2.

Step 2. Raise to the power of 2:
  (4/25)^2 = [(2/5)^2]^2 = (2/5)^4.

Step 3. Express (2/5)^4 in terms of (5/2):
  Since (2/5) is the reciprocal of (5/2), we have:
  (2/5) = (5/2)^(-1).
  Thus, (2/5)^4 = [(5/2)^(-1)]^4 = (5/2)^(-4).

Step 4. Substitute back into the equation:
  (5/2)^x = (5/2)^(-4).

Step 5. Since the bases are the same, the exponents must be equal:
  x = -4.

Thus, the solution is x = -4.