Chapter Review for Logs
Chapter
Chapter 8
Section
Chapter Review for Logs
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Solutions 48 Videos

Determine the inverse of the function. Express your answers in log form.

\displaystyle y = 4^x

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Q1a

Determine the inverse of the function. Express your answers in log form.

\displaystyle y = a^x

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Q1b

Determine the inverse of the function. Express your answers in log form.

\displaystyle y = (\frac{3}{4})^x

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Q1c

Determine the inverse of the function. Express your answers in log form.

\displaystyle m = p^q

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Q1d

Describe the transformations that must be applied to the parent function y= \log x to obtain each of the following functions.

f(x) = - \log(2x)

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Q2a

Describe how the graphs of f(x) = \log x and g(x) = 3\log(x - 1) + 2 are similar yet different.

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Q4

Evaluate.

\displaystyle \log_7343

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Q5a

Evaluate.

\displaystyle \log_{\frac{1}{5}}25

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Q5b

Evaluate.

\displaystyle \log_{19}1

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Q5c

Evaluate.

\displaystyle \log_{4}\frac{1}{256}

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Q5d

Estimate the value to three decimal places.

\displaystyle \begin{array}{llllllll} &(a) \phantom{.} \log_353 &(b) \phantom{.} \log_4 \frac{1}{10} \\ &(c) \phantom{.} \log_6 159 &(d) \phantom{.} \log_{15} 1456 \end{array}

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Q6

Express as a single logarithm.

\displaystyle \log 5 + \log 11

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Q7a

Express as a single logarithm.

\displaystyle \log 20 - \log 4

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Q7b

Express as a single logarithm.

\displaystyle \log_56 + \log_58-\log_512

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Q7c

Express as a single logarithm.

\displaystyle 2\log 3 + 4\log 2

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Q7d

Evaluate using log laws.

\displaystyle \log_642 - \log_67

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Q8a

Evaluate using log laws.

\displaystyle \log_35 + \log_318 -\log_310

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Q8b

Evaluate using log laws.

\displaystyle \log_7 \sqrt[3]{49}

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Q8c

Evaluate using log laws.

\displaystyle 2 \log_48

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Q8d

Describe how the graph of y = \log(10 000x) is related to the graph of y = \log x.

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Q9

Solve for x.

\displaystyle 5^x = 3125

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Q10a

Solve for x.

\displaystyle 4^x = 16 \sqrt{128}

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Q10b

Solve for x.

\displaystyle 4^{5x} =16^{2x-1}

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Q10c

Solve for x.

\displaystyle 3^{5x} 9^{x^2} = 27

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Q10d

Solve. Express each answer to three decimal places.

\displaystyle 6^x = 78

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Q11a

Solve. Express each answer to three decimal places.

\displaystyle 5.8^x = 234

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Q11b

Solve. Express each answer to three decimal places.

\displaystyle 8\cdot 3^x = 132

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Q11c

Solve. Express each answer to three decimal places.

\displaystyle 200 \cdot (1.23^x) = 540

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Q11d

Solve.

\displaystyle 4^x + 6 \cdot 4^{-x} = 5

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Q12a

Solve.

\displaystyle 8 \cdot 5^{2x} + 8 \cdot 5^x =6

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Q12b

The half-life of a certain substance is 3.6 days. How long will it take for 20 g of the substance to decay to 7 g?

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Q13

Solve.

\displaystyle \log_5(2x - 1) =3

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Q14a

Solve.

\displaystyle \log 3x = 4

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Q14b

Solve.

\displaystyle \log_4(3x - 5) = \log_411 + \log_42

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Q14c

Solve.

\displaystyle \log(4x - 1) = \log(x + 1) + \log 2

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Q14d

Solve.

\displaystyle \log(x + 9) - \log x = 1

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Q15a

Solve.

\displaystyle \log x + \log(x -3) = 1

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Q15b

Solve.

\displaystyle \log(x - 1) + \log(x + 2) = 1

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Q15c

Solve.

\displaystyle \log\sqrt{x^2 -1} = 2

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Q15d

Recall that L = 10 \log (frac{I}{I_0}), where I is the intensity of sound in watts per square metre (W/m^2) and I_0 = 10^{12}W/m^2. Determine the intensity of a baby screaming if the noise level is 100 dB.

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Q16

What is the sound intensity in watts per square metre (W/m^2) of an engine that is rated at 82 dB?

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Q17

How many times more intense is an earthquake of magnitude 6.2 than an earthquake of magnitude 5.5?

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Q18

Pure water has a pH value of 7.0. How many times more acidic is milk, with a pH value of 6.4, than pure water?

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Q19

Does an increase in acidity from pH 4.7 to pH 2.3 result in the same change in hydrogen ion concentration as a decrease in alkalinity from 12.5 to 10.1? Explain.

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Q20

Is an exponential model appropriate for the data in the following table? If it is, determine the equation that models the data.

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Q21

The population of a town is decreasing at the rate of 1.6%/a. If the population today is 20 000, how long will it take for the population to decline to 15 000?

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Q22

The population of a town is decreasing at the rate4 of 1.6%/a. If the population today is 20 000, how long will it take for the population to decline to 15 000?

a) Calculate the average rate of growth over the entire time period.

b) Calculate the average rate of growth for the first 30 years. How does it compare with the rate of growth for the entire time period?

c) Determine an exponential model for the data.

d) Estimate the instantaneous rate of growth in

  • i) 1970
  • ii) 1990
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Q23

The following data show the number of people (in thousands) who own a DVD player in a large city or linear is best for over a period of years.

a) Determine if an exponential or linear model is best for this data.

b) Use your model to predict how many people will own a DVD player in the year 2015.

c) What assumptions did you make to make your prediction in part b)? Do you think this is reasonable? Explain.

d) Determine the average rate of change in the number of DVD players in this city between 1999 and 2002.

e) Estimate the instantaneous rate of change in the number of DVD players in this city in 2000.

f) Explain why using an exponential model to answer part b) does not make sense.

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Q24