Check all that are true. A . ln(A ^ 2) - 3 * ln(e) . 2/3 * ln(A/e); (ln(A ^ 2))/(ln(e ^ 3)) . ln((A ^ 2)/(e ^ 3)) E. ln(A ^ 2 - e ^ 3) C.

Let's evaluate the expressions to determine which statements are true. I'll assume that "C." is simply a reference point and not part of an actual expression. I'll go through each expression and simplify them:

### Expression A: \( \ln(A^2) - 3 \cdot \ln(e) \)
1. \( \ln(A^2) \) simplifies to \( 2\ln(A) \) because \( \ln(x^y) = y\ln(x) \).
2. \( \ln(e) = 1 \), so \( 3 \cdot \ln(e) = 3 \).

Thus, the expression simplifies to:
\[ 2\ln(A) - 3 \]

### Expression B: \( \frac{2}{3} \cdot \ln\left(\frac{A}{e}\right) \)
This can be broken down as:
1. \( \ln\left(\frac{A}{e}\right) = \ln(A) - \ln(e) \).
2. \( \ln(e) = 1 \), so this becomes \( \ln(A) - 1 \).

Thus, the expression simplifies to:
\[ \frac{2}{3} \cdot (\ln(A) - 1) = \frac{2}{3}\ln(A) - \frac{2}{3} \]

### Expression C: \( \frac{\ln(A^2)}{\ln(e^3)} \)
1. \( \ln(A^2) = 2\ln(A) \).
2. \( \ln(e^3) = 3\ln(e) = 3 \) because \( \ln(e) = 1 \).

Thus, the expression simplifies to:
\[ \frac{2\ln(A)}{3} \]

### Expression D: \( \ln\left(\frac{A^2}{e^3}\right) \)
This expression can be broken down as:
1. \( \ln\left(\frac{A^2}{e^3}\right) = \ln(A^2) - \ln(e^3) \).
2. \( \ln(A^2) = 2\ln(A) \) and \( \ln(e^3) = 3 \).

Thus, the expression simplifies to:
\[ 2\ln(A) - 3 \]

### Expression E: \( \ln(A^2 - e^3) \)
This is already in its simplest form. It's the natural logarithm of the difference between \( A^2 \) and \( e^3 \).

### Comparing the Expressions:
- Expression A simplifies to \( 2\ln(A) - 3 \).
- Expression B simplifies to \( \frac{2}{3}\ln(A) - \frac{2}{3} \).
- Expression C simplifies to \( \frac{2\ln(A)}{3} \).
- Expression D simplifies to \( 2\ln(A) - 3 \).
- Expression E is \( \ln(A^2 - e^3) \).

### Conclusion:
- Expression A and Expression D are identical, so both are true relative to each other.
- Expression B and Expression C are not identical to any others and cannot be directly simplified to match A or D.
- Expression E cannot be simplified further and is distinct from the others.

**Therefore, the true statements are: A and D.**

Would you like more details or have any questions?

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**Related Questions:**

1. How does the logarithmic property \(\ln(a/b) = \ln(a) - \ln(b)\) apply to logarithmic equations?
2. What are some common mistakes when simplifying logarithmic expressions?
3. How does the natural logarithm function differ from other logarithms?
4. Why is \( \ln(e) = 1 \) and how does this property simplify expressions?
5. What happens to logarithmic expressions when the argument is negative?

**Tip:** When working with logarithms, always remember that \( \ln(e) = 1 \), as this is the natural logarithm of Euler's number.

Learn how to simplify logarithmic expressions involving natural logarithms in this detailed example. The problem covers key concepts such as logarithmic properties, natural logarithms, and algebraic simplification. Suitable for students in Grades 10-12.

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