Question 1

Expand the logarithmic expression $$\log_3$ \left( \frac{x^3$ y}{3z} \right)$$ using properties of logarithms.

Accepted Answer

$$\log_3$ x^3$ + \log_3$ y - \log_3$ 3 - \log_3$ z$$

Question 2

Solve for $$x$$: $$3 \log_2$ x + 1 = 7$$

Accepted Answer

$$x = 8$$

Question 3

Rewrite $$\log_3$ 8 = x as an $$exponential expression.

Accepted Answer

$$3^x = 8$$

Question 4

Condense and simplify: $$2 \ln(xy^2$) - 3 \ln y$$

Accepted Answer

$$2 \ln x + 2 \ln y - 3 \ln y = 2 \ln x - \ln y$$

Question 5

If the initial population is 67,000 and it decreases at a rate of 1.67% per year, which exponential function models this situation?

Accepted Answer

$$P(t) = 67000 \cdot (0.9833)^t$$

Question 6

Afghanistan suffered two major earthquakes in 1998. The first had a magnitude of $$R=6.1$$ and the second $$R=6.9. $$Using the formula $$\log\left(\frac{A}{T}\right) + B$$, how many times more powerful was the second quake compared to the first, assuming $$B$$ and $$T$$ are equal?

Accepted Answer

$$10^{6.9 - 6.1}$$ = $$10^{0.8}$$ \approx 6.3$$ times

Question 7

Solve for $$x$$: $$4x + 3 = 7x. If $$there are any extraneous solutions, explain why.

Accepted Answer

$$x = 1$$; no extraneous solutions because this is a linear equation

Question 8

Solve for $$x$$: $$\log(x+2) + \log(x-1) = 4. $$Identify any extraneous solutions.

Accepted Answer

$$x = 9$$; no extraneous solutions

Question 9

Given $$P(t) = \frac{300}{1 + e$$^{4-t}$$}$$, what is the maximum number of students infected with the flu?

Accepted Answer

$$300$$

Question 10

The pH of sea water is $$7.6$$ and the pH of milk of magnesia is $$10.5. $$What is the hydrogen ion concentration of sea water?

Accepted Answer

$$[H^+] = 10$$^{-7.6}$$

Question 11

How many times greater is the hydrogen ion concentration of sea water than of milk of magnesia? (pH of sea water $$=7.6$$, pH of milk of magnesia $$=10.5)$$

Accepted Answer

$$10^{10.5 - 7.6}$$ \approx 794$$ times$

Question 12

Write the exponential function for an initial population of 67,000 increasing at a rate of 1.67% per year.

Accepted Answer

$$P(t) = 67000 \cdot (1.0167)^t$$

Question 13

Solve for $$x$$: $$\log_{\sqrt{10}} 3 = x$$

Accepted Answer

$$x = 2$$

Question 14

Evaluate $$\ln \left( \frac{1}{e^7$} \right)$$

Accepted Answer

$$-7$$

Question 15

Solve for $$x$$: $$\frac{50}{4 + e$$^{2x}$$} = 11$$

Accepted Answer

$$x = \frac{1}{2} \ln \left( \frac{50}{11} - 4 \right)$$

Question 16

For the function $$f(x) = e$$^{x^2 - 5}$$ - 2$$, what is the horizontal asymptote?

Accepted Answer

$$y = -2$$

Question 17

For the function $$f(x) = e$$^{x^2 - 5}$$ - 2$$, what is the domain?

Accepted Answer

$$(-\infty, \infty)$$

Question 18

For the function $$f(x) = e$$^{x^2 - 5}$$ - 2$$, on which interval is the function increasing?

Accepted Answer

$$(0, \infty)$$

Question 19

For the function $$f(x) = e$$^{x^2 - 5}$$ - 2$$, what is the range?

Accepted Answer

$$(-2, \infty)$$

College Algebra: Exponential, Logarithmic, and Logistic Functions Review Guidance

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