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!