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, $e^x$.