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Let's solve the given problems step-by-step.

### Problem (a)
We are given:
\[
\ln 4 = x
\]
This is asking for the equivalent exponential form. We know that:
\[
\ln a = b \quad \text{is equivalent to} \quad e^b = a
\]
So for this case:
\[
\ln 4 = x \quad \text{is equivalent to} \quad e^x = 4
\]
Thus, in the form \( e^A = B \), we have:
- \( A = x \)
- \( B = 4 \)

### Problem (b)
We are given:
\[
\ln x = 3
\]
Again, using the same rule for converting logarithms to exponential form:
\[
\ln a = b \quad \text{is equivalent to} \quad e^b = a
\]
So for this case:
\[
\ln x = 3 \quad \text{is equivalent to} \quad e^3 = x
\]
Thus, in the form \( e^C = D \), we have:
- \( C = 3 \)
- \( D = x \)

### Final Answers:
- (a) \( A = x \), \( B = 4 \)
- (b) \( C = 3 \), \( D = x \)

Would you like further clarification on these steps?

Here are five related questions:
1. What is the general relationship between natural logarithms and exponents?
2. How do you convert logarithmic expressions of different bases to exponential form?
3. What happens when we apply a logarithmic function to exponential equations?
4. Can logarithmic functions be used to solve for unknown exponents?
5. How does the graph of \( e^x \) differ from \( \ln x \)?

**Tip**: Remember that the natural logarithm function, \( \ln(x) \), is the inverse of the exponential function, \( e^x \).

Express the equation in exponential form: (a) ln 4 = x is equivalent to e^A = B. (b) ln x = 3 is equivalent to e^C = D.

Learn how to express logarithmic equations in exponential form, including examples like ln 4 = x and ln x = 3. Understand the key formulas and concepts related to logarithms and exponential functions, suitable for students in Grades 9-12.

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