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**18. Which of the following is equivalent to \(\log\left(\frac{xyz}{w}\right)\)?**
Using the properties of logarithms:
\[ \log\left(\frac{xyz}{w}\right) = \log x + \log y + \log z - \log w \]

Answer: **(A) \(\log x + 2\log y + \log z\)**

**19. What is the solution of the equation \(2^{x+14} = 16 \cdot 2^x\)?**
First, express 16 as a power of 2:
\[ 2^{x+14} = 2^4 \cdot 2^x \]
\[ 2^{x+14} = 2^{x+4} \]
Since the bases are equal, we can equate the exponents:
\[ x + 14 = x + 4 \]
\[ 14 = 4 \]

This is not possible, so none of the answers are correct.

Answer: **(E) None of these**

**20. What is the solution of the equation \(0.5 \log_3 x = 2\)?**
First, isolate \(\log_3 x\):
\[ 0.5 \log_3 x = 2 \]
\[ \log_3 x = 4 \]
Now, convert the logarithmic equation to an exponential form:
\[ x = 3^4 = 81 \]

Answer: **(C) 81**

**21. Which function does not have a graph with asymptote \(y = 0\)?**
The function \(f(x) = e^{-5x}\) has an asymptote \(y = 0\) as \(x \to \infty\).
The function \(f(x) = \log 6x\) has a vertical asymptote at \(x = 0\), not horizontal.
The function \(f(x) = (0.25)^{x+2}\) and \(f(x) = 3^x\) both have a horizontal asymptote at \(y = 0\).

Answer: **(C) \(f(x) = \log 6x\)**

**22. What is the simplified form of \(\frac{2x + 4}{x^2 - 4}\)?**
First, factor the denominator:
\[ x^2 - 4 = (x - 2)(x + 2) \]
\[ \frac{2x + 4}{(x - 2)(x + 2)} = \frac{2(x + 2)}{(x - 2)(x + 2)} \]
Cancel the common factor:
\[ \frac{2}{x - 2} \]

Answer: **(D) \(\frac{2}{x - 2}\)**

**23. The variable \(z\) varies directly with both \(x\) and \(y\). When \(x = 6\) and \(y = \frac{1}{3}\), \(z = 30\). Which equation relates \(x\), \(y\), and \(z\)?**
Since \(z\) varies directly with both \(x\) and \(y\):
\[ z = kxy \]
Substitute the given values to find \(k\):
\[ 30 = k \cdot 6 \cdot \frac{1}{3} \]
\[ 30 = 2k \]
\[ k = 15 \]
Thus:
\[ z = 15xy \]

Answer: **(C) \(z = 15xy\)**

**24. What is the quotient \(\frac{(x + 2)}{\left( \frac{x^2 - 9x - 22}{x^2 - 121} \right)}\)?**
\[ \frac{x + 2}{\frac{x^2 - 9x - 22}{x^2 - 121}} = (x + 2) \cdot \frac{x^2 - 121}{x^2 - 9x - 22} \]
Factor the expressions:
\[ x^2 - 121 = (x - 11)(x + 11) \]
\[ x^2 - 9x - 22 = (x - 11)(x + 2) \]
Cancel the common factors:
\[ (x + 2) \cdot \frac{(x - 11)(x + 11)}{(x - 11)(x + 2)} = x + 11 \]

Answer: **(A) \(x + 11\)**

**25. What are all the solutions of the equation \(\frac{-10}{x - 9} = \frac{7}{2}\)?**
Cross-multiply to solve for \(x\):
\[ -10 \cdot 2 = 7(x - 9) \]
\[ -20 = 7x - 63 \]
\[ 7x = 43 \]
\[ x = \frac{43}{7} \]

Since this value is not among the provided options, verify any algebraic steps if necessary.

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 simplify complex fractions?
2. What are the properties of logarithmic functions?
3. How do you solve equations involving direct variation?
4. Can you provide more examples of factoring quadratic expressions?
5. What are the steps to finding the inverse of a logarithmic function?

**Tip:** When solving equations involving direct variation, always identify the constant of variation first to establish the relationship between the variables.

Explore solutions to advanced high school-level problems involving logarithmic equations, exponential equations, and direct variation. Includes step-by-step solutions and explanations for challenging mathematical concepts.

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