Basic Mathematics & Logarithm
Properties of Logarithms
Grade Class 11
Question:
<p>The expression \(\log_p\!\left(\log_p\!\left(\sqrt[p]{\sqrt[p]{\cdots\sqrt[p]{p}}}\right)\right)\), where there are \(n\) radical signs, simplifies to</p>
independent of p
independent of p and of n
dependent on both p and n
positive
Step-by-Step Solution
Key Concept: Repeated p-th roots convert p into p^(1/p^n). The inner radical equals p^(1/p^n). Then log_p of it is 1/p^n. Taking another log_p gives -n, which depends only on n and is independent of p.
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. Repeated p-th roots convert p into p^(1/p^n). The inner radical equals p^(1/p^n). Then log_p of it is 1/p^n. Taking another log_p gives -n, which depends only on n and is independent of p. Trap: The second logarithm changes the form completely; stopping after the first log gives the wrong dependence. 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: A