Calculus Volume 1 · 1.5 Exponential and Logarithmic Functions

Exponential and Logarithmic Functions: practice problems

45 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

1. Exercises 229

   For the following exercises, evaluate the given exponential functions as indicated, accurate to two significant digits after the decimal.

   $f(x)=5^x$ a. $x=3$ b. $x=\frac{1}{2}$ c.

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2. Exercises 231

   For the following exercises, evaluate the given exponential functions as indicated, accurate to two significant digits after the decimal.

   $f(x)={10}^x$ a. $x=-2$ b. $x=4$ c. $x=\frac{5}{3}$

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3. Exercises 233

   For the following exercises, match the exponential equation to the correct graph.

   1. $y=4^{\text{-}x}$
   2. $y=3^{x-1}$
   3. $y=2^{x+1}$

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4. Exercises 235

   For the following exercises, match the exponential equation to the correct graph.

   1. $y=4^{\text{-}x}$
   2. $y=3^{x-1}$
   3. $y=2^{x+1}$

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5. Exercises 237

   For the following exercises, match the exponential equation to the correct graph.

   1. $y=4^{\text{-}x}$
   2. $y=3^{x-1}$
   3. $y=2^{x+1}$

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6. Exercises 239

   For the following exercises, sketch the graph of the exponential function. Determine the domain, range, and horizontal asymptote.

   $f(x)=e^x+2$

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7. Exercises 241

   For the following exercises, sketch the graph of the exponential function. Determine the domain, range, and horizontal asymptote.

   $f(x)=3^{x+1}$

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8. Exercises 243

   For the following exercises, sketch the graph of the exponential function. Determine the domain, range, and horizontal asymptote.

   $f(x)=1-2^{\text{-}x}$

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9. Exercises 245

   For the following exercises, sketch the graph of the exponential function. Determine the domain, range, and horizontal asymptote.

   $f(x)=e^{\text{-}x}-1$

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10. Exercises 247

    For the following exercises, write the equation in equivalent exponential form.

    ${\log }_{8}2=\frac{1}{3}$

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11. Exercises 249

    For the following exercises, write the equation in equivalent exponential form.

    ${\log }_{5}25=2$

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12. Exercises 251

    For the following exercises, write the equation in equivalent exponential form.

    $\ln (\frac{1}{e^3})=-3$

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13. Exercises 253

    For the following exercises, write the equation in equivalent exponential form.

    $\ln 1=0$

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14. Exercises 255

    For the following exercises, write the equation in equivalent logarithmic form.

    $4^{-2}=\frac{1}{16}$

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15. Exercises 257

    For the following exercises, write the equation in equivalent logarithmic form.

    $9^0=1$

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16. Exercises 259

    For the following exercises, write the equation in equivalent logarithmic form.

    $\sqrt[3]{64}=4$

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17. Exercises 261

    For the following exercises, write the equation in equivalent logarithmic form.

    $9^y=150$

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18. Exercises 263

    For the following exercises, write the equation in equivalent logarithmic form.

    $4^{-3\text{/}2}=0.125$

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19. Exercises 265

    For the following exercises, sketch the graph of the logarithmic function. Determine the domain, range, and vertical asymptote.

    $f(x)=\ln (x-1)$

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20. Exercises 267

    For the following exercises, sketch the graph of the logarithmic function. Determine the domain, range, and vertical asymptote.

    $f(x)=1-\ln x$

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21. Exercises 269

    For the following exercises, sketch the graph of the logarithmic function. Determine the domain, range, and vertical asymptote.

    $f(x)=\ln (x+1)$

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22. Exercises 271

    For the following exercises, use properties of logarithms to write the expressions as a sum, difference, and/or product of logarithms.

    ${\log }_3\frac{9a^3}{b}$

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23. Exercises 273

    For the following exercises, use properties of logarithms to write the expressions as a sum, difference, and/or product of logarithms.

    ${\log }_5\sqrt{125x{y}^3}$

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24. Exercises 275

    For the following exercises, use properties of logarithms to write the expressions as a sum, difference, and/or product of logarithms.

    $\ln (\frac{6}{\sqrt{e^3}})$

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25. Exercises 277

    For the following exercises, solve the exponential equation exactly.

    $e^{3x}-15=0$

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26. Exercises 279

    For the following exercises, solve the exponential equation exactly.

    $4^{x+1}-32=0$

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27. Exercises 281

    For the following exercises, solve the exponential equation exactly.

    ${10}^x=7.21$

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28. Exercises 283

    For the following exercises, solve the exponential equation exactly.

    $7^{3x-2}=11$

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29. Exercises 285

    For the following exercises, solve the logarithmic equation exactly, if possible.

    ${\log }_{5}x=-2$

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30. Exercises 287

    For the following exercises, solve the logarithmic equation exactly, if possible.

    $\log (2x-7)=0$

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31. Exercises 289

    For the following exercises, solve the logarithmic equation exactly, if possible.

    ${\log }_6(x+9)+{\log }_{6}x=2$

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32. Exercises 291

    For the following exercises, solve the logarithmic equation exactly, if possible.

    $\ln x+\ln (x-2)=\ln 4$

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33. Exercises 293

    For the following exercises, use the change-of-base formula and either base 10 or base e to evaluate the given expressions. Answer in exact form and in approximate form, rounding to four decimal places.

    ${\log }_{7}82$

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34. Exercises 295

    For the following exercises, use the change-of-base formula and either base 10 or base e to evaluate the given expressions. Answer in exact form and in approximate form, rounding to four decimal places.

    ${\log }_{0.5}211$

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35. Exercises 297

    For the following exercises, use the change-of-base formula and either base 10 or base e to evaluate the given expressions. Answer in exact form and in approximate form, rounding to four decimal places.

    ${\log }_{0.2}0.452$

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36. Exercises 299

    For the following exercises, use the change-of-base formula and either base 10 or base e to evaluate the given expressions. Answer in exact form and in approximate form, rounding to four decimal places.

    [T] The number of bacteria N in a culture after t days can be modeled by the function $N(t)=1300\cdot {(2)}^{t\text{/}4}.$ Find the number of bacteria present after 15 days.

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37. Exercises 301

    For the following exercises, use the change-of-base formula and either base 10 or base e to evaluate the given expressions. Answer in exact form and in approximate form, rounding to four decimal places.

    [T] The amount A of a $100,000 investment paying continuously and compounded for t years is given by $A(t)=100,000\cdot e^{0.055t}.$ Find the amount A accumulated in 5 years.

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38. Exercises 303

    For the following exercises, use the change-of-base formula and either base 10 or base e to evaluate the given expressions. Answer in exact form and in approximate form, rounding to four decimal places.

    [T] The concentration of hydrogen ions in a substance is denoted by $[{\text{H}}^{+}],$ measured in moles per liter. The pH of a substance is defined by the logarithmic function $\text{pH}=\text{-}\log [{\text{H}}^{+}].$ This function is used to measure the acidity of a substance. The pH of water is 7. A substance with a pH less than 7 is an acid, whereas one that has a pH of more than 7 is a base.

    1. Find the pH of the following substances. Round answers to one digit.
    2. Determine whether the substance is an acid or a base.

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39. Exercises 305

    For the following exercises, use the change-of-base formula and either base 10 or base e to evaluate the given expressions. Answer in exact form and in approximate form, rounding to four decimal places.

    [T] According to the World Bank, at the end of 2013 ($t=0$

    ) the U.S. population was 316 million and was increasing according to the following model:

    $P(t)=316e^{0.0074t},$

    where P is measured in millions of people and t is measured in years after 2013.

    1. Based on this model, what will be the population of the United States in 2020?
    2. Determine when the U.S. population will be twice what it is in 2013.

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40. Exercises 307

    For the following exercises, use the change-of-base formula and either base 10 or base e to evaluate the given expressions. Answer in exact form and in approximate form, rounding to four decimal places.

    [T] A bacterial colony grown in a lab is known to double in number in 12 hours. Suppose, initially, there are 1000 bacteria present.

    1. Use the exponential function $Q=Q_0{e}^{kt}$ to determine the value $k,$ which is the growth rate of the bacteria. Round to four decimal places.
    2. Determine approximately how long it takes for 200,000 bacteria to grow.

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41. Exercises 309

    For the following exercises, use the change-of-base formula and either base 10 or base e to evaluate the given expressions. Answer in exact form and in approximate form, rounding to four decimal places.

    [T] The 1906 earthquake in San Francisco had a magnitude of 8.3 on the Richter scale. At the same time, in Japan, an earthquake with magnitude 4.9 caused only minor damage. Approximately how much more intense was the San Francisco earthquake than the Japanese earthquake? See the definition of Richter Scale in [link] in this section.

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42. Review Exercises 311

    True or False? Justify your answer with a proof or a counterexample.

    $f\circ g=g\circ f,$ assuming f and g are functions.

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43. Review Exercises 325

    Solve the following logarithmic equations.

    ${\log }_2(x+4)=3$

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44. Review Exercises 333

    For the following problems, consider the population of Ocean City, New Jersey, which is cyclical by season.

    The population can be modeled by $P(t)=82.5-67.5\cos [(\pi \text{/}6)t],$ where $t$ is time in months $(t=0$ represents January 1) and $P$ is population (in thousands). During a year, in what intervals is the population less than 20,000? During what intervals is the population more than 140,000?

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45. Review Exercises 335

    For the following problems, consider radioactive dating. A human skeleton is found in an archeological dig. Carbon dating is implemented to determine how old the skeleton is by using the equation $y=e^{rt},$ where $y$ is the ratio of radiocarbon still present in the material, $t$ is the number of years passed, and $r=-0.0001210$ is the decay rate of radiocarbon.

    If the skeleton is expected to be 2000 years old, what percentage of radiocarbon should be present?

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