Q: Given the system of equations: $2x - 2y + 3z = -16$, $4x - 3y + 1z = -3$, $x + y - 5z = -24$, which of the following is the correct augmented matrix representation?

$\begin{bmatrix} 2 & -2 & 3 & | & -16 \\ 4 & -3 & 1 & | & -3 \\ 1 & 1 & -5 & | & -24 \end{bmatrix}$

Q: Suppose you want to purchase new appliances for $\$4000$ in 10 years. If the annual interest rate is $6.5\%$ compounded continuously, how much do you need to invest now? (Use $A = Pe^{rt}$)

$P = \dfrac{4000}{e^{0.065 \times 10}}$

Q: If $A = \begin{bmatrix} 2 & -3 \\ -1 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} -2 & 2 \\ 3 & -2 \end{bmatrix}$, what is $A + B$?

$\begin{bmatrix} 0 & -1 \\ 2 & 2 \end{bmatrix}$

Question 1

Given the function $f(x) = C a^x$ and the data points $f(0) = 2$, $f(1) = 6$, $f(2) = 18$, $f(3) = 54$, $f(4) = 162$, what is the value of $a$?

Accepted Answer

$a = 3$

Question 2

Which of the following is the correct change of base formula for logarithms?

Accepted Answer

$\log_b a = \dfrac{\log_c a}{\log_c b}$

Question 3

Solve for $x$: $2^{x+1} = 5$

Accepted Answer

$x = \log_2 5 - 1$

Question 4

Evaluate $\log_{10} (10^4 \cdot 10^2)$ without a calculator.

Accepted Answer

$6$

Question 5

What is the domain of the function $f(x) = \log_2 (x-1)$?

Accepted Answer

$x > 1$

Question 6

If you invest $8000$ in a money market account at an annual interest rate of $8.5\%$ compounded quarterly, how much will you have in the account at the end of $10$ years? Use $A = P \left(1 + \dfrac{r}{n}\right)^{nt}$.

Accepted Answer

$A = 8000 \left(1 + \dfrac{0.085}{4}\right)^{40}$

Question 7

Which of the following statements about logarithms is INCORRECT?

Accepted Answer

$\log(x) + \log(y) = \dfrac{\log(x)}{\log(y)}$

Question 8

A radioactive isotope decays according to $A(t) = A_0 \left(\dfrac{1}{2}\right)^{t/10}$, where $t$ is in years. If you start with $12$ grams, how much will remain after $25$ years?

Accepted Answer

$12 \left(\dfrac{1}{2}\right)^{2.5}$

Question 9

Solve for $x$: $\log_2 (x + 6) = 2$

Accepted Answer

$x = 2^2 - 6 = -2$

Question 10

Which of the following best describes the transformation from $f(x) = 3^x$ to $g(x) = 3^{x-2} + 1$?

Accepted Answer

Shift right 2 units and up 1 unit

Question 11

Write $2 \ln 3 + \ln 4 - \ln 6$ as a single logarithm and simplify.

Accepted Answer

$\ln\left(\dfrac{3^2 \cdot 4}{6}\right)$

Question 12

If the population of a city grows according to $P = 660,000 e^{0.017t}$, how long will it take for the population to reach $1,000,000$?

Accepted Answer

$t = \dfrac{\ln\left(\dfrac{1,000,000}{660,000}\right)}{0.017}$

Question 13

Given the system of equations: $2x - 2y + 3z = -16$, $4x - 3y + 1z = -3$, $x + y - 5z = -24$, which of the following is the correct augmented matrix representation?

Accepted Answer

$\begin{bmatrix} 2 & -2 & 3 & | & -16 \ 4 & -3 & 1 & | & -3 \ 1 & 1 & -5 & | & -24 \end{bmatrix}$

Question 14

If $f(x) = e^{x^2 - 1}$, what is the domain of $f(x)$?

Accepted Answer

All real numbers

Question 15

Suppose you want to purchase new appliances for $\$4000$ in 10 years. If the annual interest rate is $6.5\%$ compounded continuously, how much do you need to invest now? (Use $A = Pe^{rt}$)

Accepted Answer

$P = \dfrac{4000}{e^{0.065 	imes 10}}$

Question 16

Given $\log_2 8 = x$, what is the value of $x$?

Accepted Answer

$x = 3$

Question 17

Which of the following is equivalent to $\log_5 25$?

Accepted Answer

$2$

Question 18

If $A = \begin{bmatrix} 2 & -3 \ -1 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} -2 & 2 \ 3 & -2 \end{bmatrix}$, what is $A + B$?

Accepted Answer

$\begin{bmatrix} 0 & -1 \ 2 & 2 \end{bmatrix}$

Unit 4 Review: Exponential and Logarithmic Functions, Systems, and Matrices

Study Guide - Practice Questions

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