Practice Test 3 - College Algebra

20 questions. Answer them in any order, then submit to see your score and every explanation.

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1. A data fit gives $\text{pH}=a-\log([H^+])$; if $\text{pH}=7$ when $[H^+]=10^{-7}$, find $a$.
2. Which of the following correctly applies the product rule to $\log_b\!\left(\frac{x}{y}\cdot z\right)$?
3. Which expression correctly rewrites $\log(0.001)$ as an integer using base-10 logarithms?
4. Budgeting: Determine $\frac{x+9}{3}>1$ for surplus; write solution in interval notation.
5. For the rational function $f(x) = \frac{(x-1)^2(x+2)}{(x-3)(x+2)^2}$, what are the vertical asymptotes and holes?
6. The function $g(x) = -2f(\frac{x+3}{2}) + 1$ is derived from a parent function $f(x)$. If $f(2) = 4$, what is the value of $g(1)$?
7. A linear equation in the form $\frac{px+q}{r} = \frac{sx+t}{u} + v$ is solved by first finding a common denominator. If the LCD is 30, and after clearing denominators the equation becomes $6px + 6q = 5sx + 5t + 30v$, what are the possible values for the pair $(r,u)$?
8. Which property correctly simplifies $\log(AB)$ for positive $A$ and $B$?
9. Given $\log_3(x)=4$, which expression uses change of base to rewrite it with $\ln$?
10. A small business manufactures decorative boxes. The profit $P$ (in dollars) from selling $x$ boxes per week is modeled by $P(x) = -2x^2 + 160x - 1800$. If the business must sell at least 20 boxes per week to cover fixed costs, what is the maximum weekly profit the business can achieve?
11. How can $\log_b\!\left((xy)^2\right)$ be rewritten using the power rule?
12. Cost sharing: solve $\frac{x}{3}+\frac{1}{9}=\frac{4}{9}$ dollars for $x$.
13. A population of bacteria follows the model $P(t) = 1200e^{0.15t}$, where $t$ is time in hours and $P(t)$ is the population size. If the population reaches 4800 bacteria, how much additional time is needed for the population to reach 9600 bacteria?
14. Data show a chemical's concentration drops from 80 to 40 in 3 hours; which model matches $C(t)=a+b\log t$?
15. If $f(x) = x^3$ and $g(x) = \frac{1}{2}(x + 1)^3 - 4$, then $g(x)$ can be described as $f(x)$ after which transformations?
16. Solve $2(7 + 1) - 3^2$ using order of operations.
17. Dividing items: solve $\frac{x}{9}-\frac{1}{9}=\frac{2}{3}$ for $x$.
18. What is the simplified form of $\frac{\frac{1}{a} + \frac{1}{b}}{\frac{1}{a} - \frac{1}{b}}$?
19. The rational function $r(x) = \frac{4x^2 + 3x - 1}{x^3 - 2x + 5}$ has specific end behavior characteristics. Which statement correctly describes what happens to $r(x)$ as $x$ becomes very large in magnitude?
20. Which of the following is equivalent to $\frac{4}{\sqrt[3]{2x^2}}$ when the denominator is rationalized? Give your answer in simplest form.

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