Step-by-Step Solution
Key Concept: Assume log_2 7 is rational and use prime factorization. If log_2 7 = m/n, then 2^m = 7^n. That is impossible because powers of 2 and powers of 7 have different prime factorizations. Therefore log_2 7 is irrational.
Notice that the cleanest route is to simplify the structure before computing. A clever move here is to translate the logarithmic statement into a friendlier algebraic form. Assume log_2 7 is rational and use prime factorization. If log_2 7 = m/n, then 2^m = 7^n. That is impossible because powers of 2 and powers of 7 have different prime factorizations. Therefore log_2 7 is irrational. Trap: A non-integer logarithm is not automatically rational; use the contradiction argument. Now, we invoke the power of the relevant logarithmic identity, simplify carefully, and finally verify the domain so that no extraneous answer survives.
Correct Answer: C