Question 1

Which of the following is the correct application of the change of base formula to compute $$\log_7$ 8$$?

Accepted Answer

$$\log_7$ 8 = \frac{\log 8}{\log 7}$$

Question 2

A city’s population grows according to $$p(t) = 5237e$$^{0.007t}$$. How many years will it take for the population to double its initial value? Round to the nearest tenth.

Accepted Answer

$$t = \frac{\ln 2}{0.007} \approx 99.0$$ years

Question 3

Solve for $$x$$: $$3^{x-2} = 17$. Round your answer to the nearest thousandth.

Accepted Answer

$$x = \log_3$ 17 + 2 \approx 4.578$$

Question 4

Which equation would you solve to determine when a population modeled by $$y = 3e$$^{0.04x}$$ reaches 4 million?

Accepted Answer

Solve $$3e^{0.04x} = 4$

Question 5

If $$A(t) = 250e$$^{-0.197t}$$ models radioactive decay, how long will it take for the sample to decay to half its original amount? Round to the nearest hundredth.

Accepted Answer

$$t = \frac{\ln 0.5}{-0.197} \approx 3.52$$ years

Question 6

Solve for $$x in$$ $$\log(x-3) = 1 - \log x. $$Express your answer in exact form.

Accepted Answer

Question 7

A library’s book collection grows according to $$B = 8800e$$^{0.04t}$$. How many books will there be after 5 years? Round to the nearest whole number.

Accepted Answer

$$B = 8800e$$^{0.2}$$ \approx 10773$$ books

Question 8

A sample of 700 grams of radioactive substance decays according to $$A(t) = 700e$$^{-0.036t}$$. How much will remain after 10 years? Round to the nearest whole gram.

Accepted Answer

$$A(10) = 700e$$^{-0.36}$$ \approx 486$$ grams

Question 9

The amount of particulate matter left after $$n$$ filtering steps is $$P = 800(2$$)^{-0.8n}$$. What is P$ when n = 5$? Round to the nearest whole number.

Accepted Answer

$$P = 800(2$$)^{-4}$$ \approx 50$$ units

Question 10

Which equation would you solve to determine how long it takes for a population modeled by $$y = 3e$$^{0.04x}$$ to quadruple?

Accepted Answer

Solve $$3e^{0.04x} = 12$

Question 11

Solve for $$x$$: $$3^x = 23. $$Round your answer to the nearest thousandth.

Accepted Answer

$$x = \log_3$ 23 \approx 2.843$$

Question 12

If $$p(t) = 11,718e$$^{0.007t}$$, how many years will it take for the population to reach 17,577? Round to the nearest tenth.

Accepted Answer

$$t = \frac{\ln(17,577/11,718)}{0.007} \approx 56.0$$ years

Question 13

Evaluate $$\ln(4e) to $$four decimal places.

Accepted Answer

$$\ln(4e) = \ln 4 + \ln e = \ln 4 + 1 \approx 2.3863$$

Question 14

Which of the following is the correct change of base formula for $$\log_{5} 0.838$$?

Accepted Answer

$$\log_{5} 0.838 = \frac{\log 0.838}{\log 5}$$

Question 15

If $$\ln(-x) + \ln 4 = \ln(3x - 9)$$, what is the exact solution for $$x$$?

Accepted Answer

$$x = -3$$

Question 16

Evaluate $$\log 250 + \log 34 to $$four decimal places.

Accepted Answer

$$\log 250 + \log 34 = \log(250 	imes 34) = \log 8500 \approx 3.9294$$

Question 17

If $$B = 8800e$$^{0.04t}$$, what is the percentage increase in books after 5 years compared to the initial amount?

Accepted Answer

$$\frac{B(5) - B(0)}{B(0)} 	imes 100\% = \frac{10773 - 8800}{8800} 	imes 100\% \approx 22.5\%$$

Question 18

If $$A(t) = 700e$$^{-0.036t}$$, what is the decay rate per year as a percentage?

Accepted Answer

$$3.6\%$$

College Algebra Review: Logarithms and Exponential Functions

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