Question

Let \( x \) be a positive real number such that \[ x^{8 \log_5(x) - 24} = 5^{-4} \] Then the product of all possible values of \( x \) is:

• A. 125
• B. 625
• C. 25
• D. 5

Hint:

When solving logarithmic equations, use the change of base formula and simplify step by step.

Answer 1

The Correct Option is C

We are given the equation \[ x^{8 \log_5(x) - 24} = 5^{-4} \] First, take the logarithm of both sides: \[ \log(x^{8 \log_5(x) - 24}) = \log(5^{-4}) \] Using the logarithmic property \( \log(a^b) = b \log(a) \), we get: \[ (8 \log_5(x) - 24) \log(x) = -4 \log(5) \] Now express \( \log_5(x) \) in terms of \( \log(x) \) using the change of base formula: \[ \log_5(x) = \frac{\log(x)}{\log(5)} \] Substitute this into the equation and solve for \( x \). After solving, we find that the possible values of \( x \) are such that their product is \( 25 \). Thus, the product of all possible values of \( x \) is \( 25 \).