Find the value of log base 2 of (log base 5 of 625), that is log_2 (log_5 625).

Introduction / Context:
This question asks for the value of a nested logarithm, where one logarithm appears inside another. It tests the understanding of evaluating simple logs with integer powers and then using that result as the argument for another log with a different base. Recognising perfect powers of the base and using the definition of logarithms makes this problem straightforward.

Given Data / Assumptions:
- We must evaluate log_2 (log_5 625).
- All logs are real and their arguments and bases are positive and bases are not equal to 1.
- 625 is a positive number and clearly related to base 5.

Concept / Approach:
We evaluate the inner logarithm log_5 625 first, using the definition of a logarithm: log_b a is the exponent to which b must be raised to obtain a. Because 625 is a power of 5, this inner log is an integer. Then we take the outer logarithm base 2 of that integer. Again, since the resulting number is a power of 2, the outer log simplifies to an integer as well.

Step-by-Step Solution:
Start with the inner log: log_5 625. Express 625 as a power of 5. We have 625 = 5^4. By definition, log_5 (5^4) = 4. So log_5 625 = 4. Now the original expression becomes log_2 (log_5 625) = log_2 4. Express 4 as a power of 2. We have 4 = 2^2. By definition, log_2 (2^2) = 2. Therefore log_2 (log_5 625) = 2.

Verification / Alternative check:
We can check numerically using basic knowledge of exponentials. Since 5^4 = 625, the inner log is exactly 4. The outer log asks for the exponent that produces 4 when the base is 2. Because 2^2 = 4, the outer log is exactly 2. No approximations are needed, and both steps use the direct definition of the logarithm function.

Why Other Options Are Wrong:
5: This would suggest that 2^5 equals 4, which is false because 2^5 equals 32.
10: This would suggest that 2^10 equals 4, which is false because 2^10 equals 1024.
15: This would be even larger and has no relation to the small numbers 4 and 2 appearing in the simplification.

Common Pitfalls:
Some students try to apply change of base formulas immediately and involve unnecessary decimal approximations. Others misidentify 625 as 5^3 instead of 5^4 or confuse log_5 625 with log_625 5. Another error is to compute log_2 5 or log_5 2 in the wrong place, which is not needed here. Focusing on expressing 625 and 4 as simple powers of their respective bases makes the problem quick and exact.

Final Answer:
The value of log_2 (log_5 625) is 2.