Difficulty: Medium
Correct Answer: 10^(10^10)
Explanation:
Introduction / Context:Nesting logarithms requires peeling them one by one from the outside in. The key fact is log_10(A) = 0 implies A = 1. Applying this repeatedly determines the inner values stepwise until N is obtained.
Given Data / Assumptions:
Concept / Approach:Let the innermost value be t_1 = log_10 N. Then set t_2 = log_10(t_1), t_3 = log_10(t_2). The given states log_10(t_3) = 0 ⇒ t_3 = 1. Work backward to find N.
Step-by-Step Solution:
log_10(t_3) = 0 ⇒ t_3 = 1But t_3 = log_10(t_2) ⇒ log_10(t_2) = 1 ⇒ t_2 = 10t_2 = log_10(t_1) ⇒ log_10(t_1) = 10 ⇒ t_1 = 10^{10}t_1 = log_10 N ⇒ log_10 N = 10^{10} ⇒ N = 10^{(10^{10})}Verification / Alternative check:Substitute forward: log_10 N = 10^{10}; log_10(10^{10}) = 10; log_10(10) = 1; log_10(1) = 0 ✓.
Why Other Options Are Wrong:10^10 and 10^100 are intermediate inner values (t_1 or a confusion), not N; 10^1010 misreads exponent placement.
Common Pitfalls:Losing track of nesting levels or assuming log_10(0)=… (undefined). Maintain positivity of each inner argument.
Final Answer:10^(10^10)
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