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.