Q: A $$14,000$$ deposit is compounded quarterly at $$8\%$$ for $$5$$ years. What is the future value? Use $$A = P(1 + \frac{r}{n}$$)^{nt}$$.

$$A = 14000(1 + \frac{0.08}{4}$$)^{4 \times 5}$$ \approx 20,601.60$$

Q: A $$21,000$$ deposit is compounded continuously at $$7.25\%$$ for $$12$$ years. What is the future value? Use $$A = Pe$$^{rt}$$.

$$A = 21000 \times e$$^{0.0725 \times 12}$$ \approx 51,210.00$$

Q: The value of a Porsche 928 increased from $$8,000 in 2000 to$$ $$15,400 in 2015. $$Assuming exponential growth, what is the exponential growth function $$V(t)$$, where $$t is $$years after 2000?

$$V(t) = 8000 \times e$$^{0.0456t}$$

Question 1

Which of the following sets represents a one-to-one function?

Accepted Answer

{(10, 5), (6, -3), (7, 4), (-2, -11)}

Question 2

What is the inverse of the function $$y = \frac{5x - 3}{2}$$?

Accepted Answer

$$y = \frac{2x + 3}{5}$$

Question 3

A $$14,000$$ deposit is compounded quarterly at $$8\%$$ for $$5$$ years. What is the future value? Use $$A = P(1 + \frac{r}{n}$$)^{nt}$$.

Accepted Answer

$$A = 14000(1 + \frac{0.08}{4}$$)^{4 	imes 5}$$ \approx 20,601.60$$

Question 4

Simplify $$\log_2$ 16$$ without using the change of base formula.

Accepted Answer

$$4$$

Question 5

Using the change of base formula, what is $$\log_4$ 23$$ rounded to the nearest thousandth?

Accepted Answer

$$2.396$$

Question 6

Express $$\ln 2x + 3 \ln x as a $$single logarithm and simplify.

Accepted Answer

$$\ln(2x^3$)$$

Question 7

Given $$\log_b 3 = 0.415$$, $$\log_b 5 = 0.568$$, and $$\log_b 17 = 1.873$$, what is $$\log_b 15$$?

Accepted Answer

$$0.983$$

Question 8

Solve the exponential equation $$54x - 7 = 125. $$What is the value of $$x$$?

Accepted Answer

$$x = 3$$

Question 9

Solve the logarithmic equation $$\log_4$ (x + 3) + \log_4$ (x - 3) = 2. Is $$the solution valid or extraneous?

Accepted Answer

Valid, $$x = 5$$

Question 10

A $$21,000$$ deposit is compounded continuously at $$7.25\%$$ for $$12$$ years. What is the future value? Use $$A = Pe$$^{rt}$$.

Accepted Answer

$$A = 21000 	imes e$$^{0.0725 	imes 12}$$ \approx 51,210.00$$

Question 11

Simplify $$\ln e^3$.

Accepted Answer

$$3$$

Question 12

Express $$\ln x + \ln(x + 3) - \ln(x - 5) as a $$single logarithm.

Accepted Answer

$$\ln \left( \frac{x(x + 3)}{x - 5} \right)$$

Question 13

If $$\log_b 3 = 0.415$$, $$\log_b 5 = 0.568$$, and $$\log_b 17 = 1.873$$, what is $$\log_b \frac{5}{17}$$ rounded to the nearest thousandth?

Accepted Answer

$$-1.305$$

Question 14

Solve the logarithmic equation $$\log(x + 5) - \log(x - 3) = \log 2. Is $$the solution valid or extraneous?

Accepted Answer

Valid, $$x = 1$$

Question 15

The value of a Porsche 928 increased from $$8,000 in 2000 to$$ $$15,400 in 2015. $$Assuming exponential growth, what is the exponential growth function $$V(t)$$, where $$t is $$years after 2000?

Accepted Answer

$$V(t) = 8000 	imes e$$^{0.0456t}$$

Question 16

Estimate the value of the Porsche 928 in 2011 using the exponential growth function $$V(t) = 8000 	imes e$$^{0.0456t}$$.

Accepted Answer

$$V(11) \approx 12,800$$

Question 17

What is the doubling time for the value of the Porsche 928, given the exponential growth function $$V(t) = 8000 	imes e$$^{0.0456t}$$?

Accepted Answer

$$t = \frac{\ln 2}{0.0456} \approx 15.2$$ years

Question 18

After how many years will the Porsche 928 be worth $$25,000$$, assuming no change in growth rate ($$V(t) = 8000 	imes e$$^{0.0456t}$$)?

Accepted Answer

$$t = \frac{\ln(25000/8000)}{0.0456} \approx 24.2$$ years

Question 19

Simplify $$\log_8$ 64$$ without using the change of base formula.

Accepted Answer

$$2$$

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