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Let's solve the given logarithmic and exponential expressions from the homework.

### (a) \( \ln(e^7) \)

Using the property of logarithms, \( \ln(e^x) = x \), we have:
\[
\ln(e^7) = 7
\]

### (b) \( e^{\ln(4)} \)

Using the inverse property \( e^{\ln(x)} = x \), we have:
\[
e^{\ln(4)} = 4
\]

### (c) \( e^{\ln(\sqrt{4})} \)

First simplify \( \sqrt{4} = 2 \), so the expression becomes:
\[
e^{\ln(2)} = 2
\]

### (d) \( \ln\left(\frac{1}{e^4}\right) \)

We can rewrite this expression as:
\[
\ln\left(e^{-4}\right)
\]
Using the logarithmic property \( \ln(e^x) = x \), we have:
\[
\ln\left(e^{-4}\right) = -4
\]

### Summary of answers:
- (a) \( \ln(e^7) = 7 \)
- (b) \( e^{\ln(4)} = 4 \)
- (c) \( e^{\ln(\sqrt{4})} = 2 \)
- (d) \( \ln\left(\frac{1}{e^4}\right) = -4 \)

Would you like a deeper explanation of any of these steps?

Here are 5 related questions to consider:
1. How do logarithmic and exponential functions relate to each other?
2. What are some common properties of logarithms that simplify calculations?
3. Why does \( \ln(e^x) = x \) hold for any real number \( x \)?
4. What is the domain of the natural logarithm function?
5. How can we apply logarithmic properties to solve real-world problems?

**Tip**: Always look for properties like \( \ln(e^x) = x \) and \( e^{\ln(x)} = x \) to simplify expressions quickly!

Find the exact value of each expression: (a) ln(e^7), (b) e^(ln(4)), (c) e^(ln(√4)), (d) ln(1/e^4).

Learn how to solve expressions involving logarithms and exponentials, such as ln(e^7), e^(ln(4)), and ln(1/e^4). This step-by-step guide explains key concepts and formulas, ideal for high school students.

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