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

$$x \u003e 1$$

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

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