Equation: 5^(x-2) * 3^(2x-3) = 135

To find the value of x in the given equation, we will follow these steps:

Step 1: Simplify the equation

To simplify the equation, we need to express the number 135 as a product of powers of 5 and 3.

The prime factorization of 135 is: 3^3 * 5

So, we can rewrite the equation as:
5^(x-2) * 3^(2x-3) = 3^3 * 5

Step 2: Apply the exponent rules

Using the exponent rules, we can simplify the equation further.

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

Rewriting 3^3 as 3^(2x-3):
5^(x-2) * 3^(2x-3) = 3^(2x-3) * 5

Step 3: Equate the exponents

Since the bases are the same, we can equate the exponents.

x - 2 = 2x - 3

Step 4: Solve for x

By rearranging the equation, we can solve for x.

x - 2x = -3 + 2

-x = -1

x = 1

Step 5: Check the solution

To verify if x = 1 is the correct solution, substitute it back into the original equation.

5^(x-2) * 3^(2x-3) = 135

5^(1-2) * 3^(2(1)-3) = 135

5^(-1) * 3^(-1) = 135

1/5 * 1/3 = 135

1/15 = 135

Since 1/15 is not equal to 135, the solution x = 1 is incorrect.

Conclusion: There is no solution for the given equation.