Let's simplify the given mathematical expression:

$\log_4(\log_2 4) - \log_5(\log_7 7)$

Step 1: Simplify each logarithmic component.

First term: $\log_4(\log_2 4)$

• $\log_2 4 = 2$ because $2^2 = 4$.
• Substituting, this becomes $\log_4(2)$.
• Using the change of base formula, $\log_4(2) = \frac{\log 2}{\log 4}$.
• Since $\log 4 = 2\log 2$, we have: $\log_4(2) = \frac{\log 2}{2\log 2} = \frac{1}{2}.$

Second term: $\log_5(\log_7 7)$

• $\log_7 7 = 1$ because $7^1 = 7$.
• Substituting, this becomes $\log_5(1)$.
• $\log_5(1) = 0$ because any logarithm of 1 is 0.

Step 2: Substitute the simplified values into the expression.

$\log_4(\log_2 4) - \log_5(\log_7 7) = \frac{1}{2} - 0 = \frac{1}{2}.$

Final Answer:

$\boxed{\frac{1}{2}}$

Would you like additional explanations or related practice problems? Here are some related questions:

1. What is the change of base formula and how is it applied in logarithmic problems?
2. Can you solve $\log_3(\log_9 81)$ step by step?
3. How do properties of logarithms simplify multi-step problems?
4. Why is $\log_b(1) = 0$ for any base $b$?
5. How does $\log_a a^k = k$ apply in logarithmic evaluations?

Tip: Always simplify inner logarithmic terms first when solving nested logarithms!