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**09. If ln(e^2x) = 1 and ln(y^2) = 2, what is x + y?**
Given:
\[ \ln(e^{2x}) = 1 \implies 2x = 1 \implies x = \frac{1}{2} \]
\[ \ln(y^2) = 2 \implies y^2 = e^2 \implies y = e \text{ or } y = -e \]
\[ x + y = \frac{1}{2} + e \text{ or } x + y = \frac{1}{2} - e \]

The correct answer is not provided in the options. Answer: **(E) None of these**

**10. Use change of base to evaluate \(\log_8 16\):**
\[ \log_8 16 = \frac{\log_{10} 16}{\log_{10} 8} = \frac{\log_2 16}{\log_2 8} = \frac{4}{3} \]

Answer: **(C) \(\frac{4}{3}\)**

**11. What is the domain of \( f(x) = \tan x \)?**
The domain of \( \tan x \) is all real numbers except where \( \cos x = 0 \), which occurs at:
\[ x \neq \frac{\pi}{2} + n\pi, n \in \mathbb{Z} \]

Answer: **(D) \( x \neq \frac{\pi}{2} + n\pi \)**

**12. What is the range of \( f(x) = \sin 2x \)?**
The range of \( \sin 2x \) is from -1 to 1.

Answer: **(B) \(-1 \leq y \leq 1\)**

**13. What is the sum of the period and the amplitude of \( y = -5 \sin(\pi(x - 2)) + 4 \)?**
- Amplitude: \( | -5 | = 5 \)
- Period: \( \frac{2\pi}{\pi} = 2 \)
Sum: \( 5 + 2 = 7 \)

Answer: **(A) 7**

**14. What is the area of an equilateral triangle with sides of length 4?**
The area \( A \) of an equilateral triangle with side length \( a \) is given by:
\[ A = \frac{\sqrt{3}}{4} a^2 = \frac{\sqrt{3}}{4} \cdot 4^2 = 4\sqrt{3} \]

Answer: **(B) \( 4\sqrt{3} \)**

**15. Suppose you deposit money in an investment account that pays 7% annual interest compounded continuously. About how many years will it take for your initial deposit to double? Use \( \ln 2 \approx 0.7 \).**
The formula for continuous compounding is \( A = Pe^{rt} \).
\[ 2 = e^{0.07t} \]
\[ \ln 2 = 0.07t \]
\[ t = \frac{0.7}{0.07} = 10 \]

Answer: **(D) 10**

**16. Which function is the inverse of \( y = \ln(x - 2) \)?**
\[ y = \ln(x - 2) \implies x - 2 = e^y \implies x = e^y + 2 \]

Answer: **(A) \( y = e^x + 2 \)**

**17. Which of the following is equivalent to \( \log_2 7 \)?**
\[ \log_2 7 = \frac{\log 7}{\log 2} \]

Answer: **(C) \( 7 \log 2 \)**

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Would you like any further details on any of the solutions? Here are five more questions you could ask:

1. Can you explain how to solve logarithmic equations in general?
2. What are some common properties of trigonometric functions?
3. How do you find the inverse of a function?
4. What is the significance of continuous compounding in finance?
5. Can you provide more examples of solving for the domain and range of functions?

**Tip:** When solving for the domain and range of trigonometric functions, it's essential to consider the values for which the function is undefined and the possible outputs of the function.

Explore solutions to mathematical problems involving logarithms, trigonometric functions, and continuous compounding. Learn about change of base, domain and range of trigonometric functions, and continuous compounding in finance. Suitable for high school students.

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