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The problem is asking to convert exponential equations into logarithmic form. Let's go through each part.

### Part (a)
The given equation is:
\[
4^5 = 1024
\]
In logarithmic form, an equation \(a^x = b\) can be written as \(\log_a b = x\).

Applying this to \(4^5 = 1024\):
\[
\log_4 1024 = 5
\]
So:
- \(A = 1024\)
- \(B = 5\)

### Part (b)
The given equation is:
\[
10^{-2} = 0.01
\]
Applying the same logic, we can rewrite \(10^{-2} = 0.01\) as:
\[
\log_{10} 0.01 = -2
\]
So:
- \(C = 0.01\)
- \(D = -2\)

### Summary of Answers
- **(a)** \( A = 1024 \), \( B = 5 \)
- **(b)** \( C = 0.01 \), \( D = -2 \)

Would you like further details on logarithmic conversions or have any questions?

### Related Questions
1. How do you convert any exponential equation into logarithmic form?
2. What is the general rule for changing the base in logarithmic expressions?
3. Why is \(\log_{10} 0.01 = -2\)? How does the negative exponent affect the calculation?
4. What are other ways to simplify logarithmic expressions?
5. How does understanding logarithmic form help in solving exponential growth problems?

**Tip:** Remember that logarithms answer the question, "To what power must the base be raised, to yield a specific number?"

Express the equation in logarithmic form: (a) 4^5 = 1024 is equivalent to log_4 A = B. (b) 10^-2 = 0.01 is equivalent to log_10 C = D.

Learn how to convert exponential equations to logarithmic form with examples like 4^5 = 1024 and 10^-2 = 0.01. This problem covers key concepts in algebra, exponential, and logarithmic functions. Suitable for high school students in Grades 10-12.

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