Difficulty: Easy
Correct Answer: 16
Explanation:
Introduction / Context:Powers of logs often simplify via base-matching identities. Here 25 is a power of 5, and the exponent involves log base 5, which suggests rewriting to use a^log_a(b) = b.
Given Data / Assumptions:Compute 25^{log_5 4}, noting 25 = 5^2.
Concept / Approach:Use exponent rules: (5^2)^{log_5 4} = 5^{2 * log_5 4}. Then apply the identity 5^{log_5 4} = 4 to simplify.
Step-by-Step Solution:
25^{log_5 4} = (5^2)^{log_5 4}= 5^{2 * log_5 4}= (5^{log_5 4})^2 = 4^2 = 16Verification / Alternative check:Numerically, log_5 4 ≈ 0.861... Then 25^{0.861...} ≈ 16 ✓.
Why Other Options Are Wrong:5 or 25 result from misapplying a^log_a(b) = b or forgetting the outer square.
Common Pitfalls:Applying log identity directly to 25 rather than rewriting it as 5^2 first.
Final Answer:16
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