Evaluate the logarithmic expression: Find the numerical value of log base 2 of (log base 5 of 625), that is, compute log base 2 (log base 5 625).

Introduction:
This problem checks your understanding of nested logarithms, where one logarithm expression appears inside another. Correctly evaluating such expressions requires applying basic logarithm rules in the right order and simplifying step by step.

Given Data / Assumptions:

• We must find the value of log base 2 (log base 5 625).
• All logarithms are assumed to have positive arguments and valid bases (not equal to 1).
• 625 is a power of 5, which will simplify the inner logarithm.

Concept / Approach:
To solve nested logarithms, first evaluate the inner logarithm completely, then treat its result as a regular number when evaluating the outer logarithm. Here, the inner term log base 5 of 625 can be simplified by expressing 625 as a power of 5. Once that is done, the outer log base 2 becomes straightforward because we are left with the logarithm of an integer.

Step-by-Step Solution:
Step 1: Simplify the inner logarithm log base 5 625. Observe that 625 = 5^4. Therefore, log base 5 625 = log base 5 (5^4) = 4. Step 2: Substitute this value into the outer logarithm. The original expression becomes log base 2 (4). Note that 4 = 2^2. Therefore, log base 2 (4) = log base 2 (2^2) = 2. Thus, the value of log base 2 (log base 5 625) is 2.

Verification / Alternative check:
We can verify by using the definition of a logarithm. Since 5^4 = 625, the inner log is exactly 4. Since 2^2 = 4, the outer log base 2 of 4 is exactly 2. No approximations are involved, so the result is exact and clearly correct.

Why Other Options Are Wrong:
The options 5, 10, and 15 would require either 625 or 4 to be a very different power of 5 or 2, which is not true. For example, log base 2 (4) is definitely not 5 or 10. These incorrect values usually arise from confusing the base and argument or from skipping steps when simplifying powers.

Common Pitfalls:
Common mistakes include trying to directly combine the logs without first evaluating the inner one, or thinking that log base 2 (log base 5 625) somehow equals log base 10 of 625. Remember that the base of a logarithm matters, and each log must be processed with its own base. Always simplify from the inside out when functions are nested.

Final Answer:
The value of log base 2 (log base 5 625) is 2.