Final answer:
To solve 5^(5x-2) = 32, convert 32 to a power of 5 and set the exponents equal. Solve the resulting equation algebraically to find x.
Step-by-step explanation:
The equation to solve is 55x-2 = 32. First, recognize that 32 is a power of 2, specifically 25. Since 5 is a prime number, it cannot be easily converted to a base of 2, so we approach the problem by converting 32 to a base of 5.
32 in base 5 can be calculated using trial and error, approximation, or by recognizing that 32 is close to 53 which is 125. Through approximation, we can establish that 32 is between 52 (which is 25) and 53 (which is 125). The exact value, 25 or 32, can be expressed as 5 to the power of log532. So, the equation becomes 55x-2 = 5log532.
Since the bases match, the exponents must be equal. So we set 5x - 2 = log532. Solving for x gives us:
- 5x = log532 + 2
- x = (log532 + 2) / 5
Now, x can be found using a calculator.