<p>\(\log_{10}(\log_2 3) + \log_{10}(\log_3 4) + \log_{10}(\log_4 5) + \cdots + \log_{10}(\log_{1023} 1024)\) simplifies to</p>
Step-by-Step Solution
Key Concept: Use log A + log B = log(AB) and then telescope the product. The sum is log_10[(log_2 3)(log_3 4)...(log_1023 1024)]. The product telescopes to log_2 1024 = 10, so the sum is log_10 10 = 1, an integer.
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. Use log A + log B = log(AB) and then telescope the product. The sum is log_10[(log_2 3)(log_3 4)...(log_1023 1024)]. The product telescopes to log_2 1024 = 10, so the sum is log_10 10 = 1, an integer. Trap: The telescoping happens inside the product of inner logarithms, not directly in the original sum. 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: D