LHS AP Precalculus Unit Review 2.9-2.13 KEY PDF

# LHS AP Precalculus Unit 2.9-2.13 Review ## Multiple Choice Questions - NO Calculators are allowed! 1. What is the value of the expression $log_4(16) - log_3(27)$? - (A) 7 - (B) 5 - (C) -1 - (D) 5/2 2. Let $w$ and $z$ be positive constants. Which of the following is equivalent t...

# LHS AP Precalculus Unit 2.9-2.13 Review ## Multiple Choice Questions - NO Calculators are allowed! 1. What is the value of the expression $log_4(16) - log_3(27)$? - (A) 7 - (B) 5 - (C) -1 - (D) 5/2 2. Let $w$ and $z$ be positive constants. Which of the following is equivalent to $log_4(16wz^3)$? - (A) 16 + $log_4(w)$ + 3$log_4(z)$ - (B) 4 + $log_4(w)$ + 3$log_4(z)$ - (C) 2 + $log_4(w)$ + 3$log_4(z)$ - (D) 16 + $log_4(w)$ - 3$log_4(z)$ 3. A function $f$ has the end behavior described by the statements $lim_{x \to 0^+} f(x) = \infty$ and $lim_{x \to \infty} f(x) = -\infty$. Which of the following could be an expression for $f(x)$? - (A) 3($\frac{1}{2}$)^x - (B) -3(2)^x - (C) 4$log_{10}(x)$ - (D) -4$log_{10}(x)$ 4. The logarithmic function $l$ is given by $l(x) = a ln(x)$, where $a$ is a constant. If $a>0$, which of the following statements about $l$ is correct? - (A) $l$ is increasing and concave up. - (B) $l$ is increasing and concave down. - (C) $l$ is decreasing and concave up. - (D) $l$ is decreasing and concave down. 5. Selected values of the function $g$ are shown in the table. Which of the following claim and explanation statements best fits these data? | x | g(x) | |---|---| | 1 | -81 | | 2 | -54 | | 3 | -36 | | 4 | -24 | | 5 | -16 | - (A) g is best modeled by a logarithmic function because the outputs over consecutive equal-length input-value intervals are proportional. - (B) g is best modeled by a logarithmic function because the inputs over consecutive equal-length output-value intervals are proportional. - (C) g is best modeled by an exponential function because the outputs over consecutive equal-length input-value intervals are proportional. - (D) g is best modeled by an exponential function because the inputs over consecutive equal-length output-value intervals are proportional. 6. The graph of the function $g$ is shown above and has a vertical asymptote at $x=0$. Which of the following claims is correct? - (A) The function $g$ could be $g(x) = -log_2(x)$ and $lim_{x \to 0^+} g(x) = -\infty$. - (B) The function $g$ could be $g(x) = log_2(x)$ and $lim_{x \to 0^+} g(x) = -\infty$. - (C) The function $g$ could be $g(x) = -log_2(x)$ and $lim_{x \to \infty} g(x) = 0$. - (D) The function $g$ could be $g(x) = log_2(x)$ and $lim_{x \to \infty} g(x) = 0$. 7. The logarithmic function $g$ is given by $g(x) = ln(x)$. Which of the following statements about $g$ is correct? - (A) $g$ is increasing and the rate of change of $g$ is increasing. - (B) $g$ is increasing and the rate of change of $g$ is decreasing. - (C) $g$ is decreasing and the rate of change of $g$ is increasing. - (D) $g$ is decreasing and the rate of change of $g$ is decreasing. 8. The graph of the function $k$ is shown above. Which of the following statements about $k$ is correct? - (A) The rate of change of $k$ is positive and increasing. - (B) The rate of change of $k$ is positive and decreasing. - (C) The rate of change of $k$ is negative and increasing. - (D) The rate of change of $k$ is negative and decreasing. 9. Let $g(x) = -3log_2(x + 4) + 1$. Which of the following gives the domain of the function $g$? - (A) $(0, \infty)$ - (B) $(-4, \infty)$ - (C) $(4, \infty)$ - (D) $(1, \infty)$ - (E) $(-\infty,0)$ 10. The graph of $f$ is shown above. Which of the following could be the equation for $f$? - (A) $f(x) = 2log_4(x)$ - (B) $f(x) = -2log_4(x)$ - (C) $f(x) = log_2(x)$ - (D) $f(x) = (\frac{1}{2})^x$ 11. The graph of $f$ is shown above. Which of the following statements about $f$ is correct? - (A) The function $f$ is decreasing and the rate of change of $f$ is positive. - (B) The function $f$ is decreasing and the rate of change of $f$ is negative. - (C) The function f is increasing and the rate of change of $f$ is increasing. - (D) The function $f$ is increasing and the rate of change of $f$ is decreasing. 12. The logarithmic function $k$ is defined by $k(x) = -2ln(x)$. Which of the following statements about $k$ is correct? - (A) The function $k$ is increasing and the graph of $k$ is concave up. - (B) The function $k$ is increasing and the graph of $k$ is concave down. - (C) The function $k$ is decreasing and the graph of $k$ is concave up. - (D) The function $k$ is decreasing and the graph of $k$ is concave down. 13. Which of the following expressions is equivalent to $ln(\frac{2x^3}{y^\frac{1}{2}})$, where $x$ and $y$ are positive constants? - (A) $ln(2) + 3ln(x) -2ln(y)$ - (B) $ln(2) + 3ln(x) - \frac{1}{2}ln(y)$ - (C) $3ln(2x)-ln(y)$ - (D) $ln(2) + 3ln(x) + \frac{1}{2}ln(y)$ 14. Which of the following is equivalent to $4log(x)-log(4) + \frac{1}{2}log(y)$, where $x$ and $y$ are positive constants? - (A) $log(\frac{x^4}{4\sqrt{y}})$ - (B) $log(\frac{x^4\sqrt{y}}{4})$ - (C) $log(\frac{x^4}{4y})$ - (D) $log(\frac{x^4\sqrt{y}}{4y})$ 15. The graph of the function $f$ is shown above. Which of the following statements about $f$ is correct? - (A) $f$ is increasing at an increasing rate. - (B) $f$ is increasing at a decreasing rate. - (C) $f$ is decreasing at an increasing rate. - (D) $f$ is decreasing at a decreasing rate. 16. The logarithmic function $h$ is defined by $h(x) = 3ln(x)$. Which of the following statements about $h$ is correct? - (A) The function $h$ is increasing and the graph of $h$ is concave up. - (B) The function $h$ is increasing and the graph of $h$ is concave down. - (C) The function $h$ is decreasing and the graph of $h$ is concave up. - (D) The function $h$ is decreasing and the graph of $h$ is concave down. 17. The graph of $f$ is shown above. Which of the following could be the equation for $f$? - (A) $f(x) = 2log_4(x)$ - (B) $f(x) = -2log_4(x)$ - (C) $f(x) = -2(\frac{1}{3})^x$ - (D) $f(x) = -2(3)^x$ 18. The logarithmic function $l$ is defined by $l(x) = -3log(x)$. Which of the following pairs of statements correctly describes the end behavior of $l$? - (A) $lim_{x \to 0^+} l(x) = -\infty$ and $lim_{x \to \infty} l(x) = \infty$ - (B) $lim_{x \to 0^+} l(x) = \infty$ and $lim_{x \to \infty} l(x) = -\infty$ - (C) $lim_{x \to 0^+} l(x) = 0$ and $lim_{x \to \infty} l(x) = \infty$ - (D) $lim_{x \to 0^+} l(x) = -\infty$ and $lim_{x \to \infty} l(x) = 0$ 19. The logarithmic function $L$ can be expressed as $L(x) = Alog_3(x)$ and also as $L(x) = Blog_2(x)$. Which of the following is an expression for A in terms of B? - (A) $A = Blog_2(3)$ - (B) $A = \frac{B}{log_3(2)}$ - (C) $A = \frac{B}{log_2(3)}$ - (D) $A = \frac{log_2(3)}{log_3(2)}$ 20. The functions $f$ and $g$ are given by $f(x) = log_3(x-2)$ and $g(x) = log_3(2x-1)$. Which of the following is all $x$ for which $f(x) + g(x) > 2$? - (A) $x>3.5$ only - (B) $x < -1$ only - (C) $x > 3.5 $ and $x < -1$ - (D) $-1 < x < 3.5$ 21. A student is solving the equation $5^x = 10^{3x-1}$. They do not have their calculator, so they must rely on a table of logarithms which provides values of $log_{10}(x)$ for various $x$. Solving which of the following equations would enable the student to solve the original equation? - (A) $x = 3x-1$ - (B) $xlog_{10}(5) = 3x-1$ - (C) $xlog_{10}(5) = 3x-1$ - (D) $x + log_{10}(5) = 3x - 1$ 22. Which of the following gives all values of $x$ for which $ln(x) + ln(x+2)=ln(24)$? - (A) $x = -6$ only - (B) $x = 4$ only - (C) $x = -6$ and $x = 4$ - (D) $x = 4$ and $x = 11$ 23. What is the value of the expression $ln(e^2) - ln(e^\frac{3}{2})$? - (A) $e^2$ - (B) $\frac{1}{2}$ - (C) $e^\frac{5}{2}$ - (D) $\frac{5}{2}$ ## Free Response Questions - NO Calculators are allowed! ### FRQ 1 The functions $f$ and $g$ are given by $$f(x) = 4(2)^{x-1} + 3$$ and $$g(x) = 5log(x + 2) + 1$$ (A) Find $f^{-1}(x)$. (B) Find $g^{-1}(x)$. (C) Let $h(x) = 32^{x+1} + 5$ and let $k(x) = h^{-1}(x)$. For what value(s) of $x$ does $k(x) = 1$? ### FRQ 2 The functions $f$ and $g$ are given by $$f(x) = log_3(x - 4)$$ and $$g(x) = 2ln(x).$$ (A) Solve $f(x) = 2$ for values of $x$ in the domain of $f$. (B) Solve $g(x) = 10$ for values of $x$ in the domain of $g$. ### FRQ 3 The functions $j$ and $k$ are given by $$j(x) = 4ln(x) -\frac{1}{2}ln(y)$$ and $$k(x) = log_{10}(x + 5) - 3log_{10}(x - 2) - log_{10}(4).$$ (A) Rewrite $j(x)$ as a single natural logarithm without negative exponents in any part of the expression. Your results should be of the form $ln(expression)$. (B) Rewrite $k(x)$ as a single logarithm base 10 without negative exponents in any part of the expression. Your result should be of the form $log_{10}(expression)$. (C) Consider the functions $f$ and $g$ given by $f(x) = log_{10}(x) + log_{10}(x+4)$ and $g(x) = log_{10}(x+18)$. In the xy-plane, what are all x-coordinates of the points of intersection of the graphs of $f$ and $g$? (D) What are all values of $x$ for which $3^{2x}=9$? (E) The function $k$ is given by $k(x) = 6e^{x+2}$. Solve $k(x) = 11$ for values of $x$ in the domain of $k$. ### FRQ 4 (A) The functions $f$ and $g$ are given by $$f(x) = 2.3(2^{x+1})$$ and $$g(x) = log_{10}(x^3) + log_{10}(\sqrt{x}).$$ (i) Rewrite $f(x)$ as an expression of the form $f(x) = ab^x$. (ii) Rewrite $g(x)$ as a single base 10 logarithmic of $x$. Your answer should be of the form $alog_{10}(x)$. (B) The functions $h$ and $j$ are given by $$h(x) = 2e^{(x+3)} - 8$$ and $$j(x) = 2ln(x) - 1.$$ (i) Solve $h(x) = 0$. (ii) Solve $j(x) = 5$ for values of $x$ in the domain of $j$.