Exponential and Logarithmic Equations and Models: Get Stronger

Logarithmic Properties

For the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.

  1. [latex]\log_b(7x \cdot 2y)[/latex]
  2. [latex]\log_b\left(\dfrac{13}{17}\right)[/latex]
  3. [latex]\ln\left(\dfrac{1}{4^k}\right)[/latex]

For the following exercises, condense to a single logarithm if possible.

  1. [latex]\ln(7) + \ln(x) + \ln(y)[/latex]
  2. [latex]\log_b(28) - \log_b(7)[/latex]
  3. [latex]-\log_b\left(\dfrac{1}{7}\right)[/latex]

For the following exercises, use the properties of logarithms to expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.

  1. [latex]\log\left(\dfrac{x^{15}y^{13}}{z^{19}}\right)[/latex]
  2. [latex]\log\left(\sqrt{x^3y^{-4}}\right)[/latex]
  3. [latex]\log\left(x^2y^3\sqrt[3]{x^2y^5}\right)[/latex]

For the following exercises, condense each expression to a single logarithm using the properties of logarithms.

  1. [latex]\ln(6x^9) - \ln(3x^2)[/latex]
  2. [latex]\log(x) - \dfrac{1}{2}\log(y) + 3\log(z)[/latex]

For the following exercise, rewrite each expression as an equivalent ratio of logs using the indicated base.

  1. [latex]\log_7(15)[/latex] to base [latex]e[/latex]

For the following exercises, suppose [latex]\log_5(6) = a[/latex] and [latex]\log_5(11) = b[/latex]. Use the change-of-base formula along with properties of logarithms to rewrite each expression in terms of [latex]a[/latex] and [latex]b[/latex]. Show the steps for solving.

  1. [latex]\log_{11}(5)[/latex]
  2. [latex]\log_{11}\left(\dfrac{6}{11}\right)[/latex]

For the following exercises, use properties of logarithms to evaluate without using a calculator.

  1. [latex]6\log_8(2) + \dfrac{\log_8(64)}{3\log_8(4)}[/latex]

For the following exercises, use the change-of-base formula to evaluate each expression as a quotient of natural logs. Use a calculator to approximate each to five decimal places.

  1. [latex]\log_3(22)[/latex]
  2. [latex]\log_6(5.38)[/latex]
  3. [latex]\log_{\frac{1}{2}}(4.7)[/latex]

Exponential and Logarithmic Equations and Models

For the following exercises, use like bases to solve the exponential equation.

  1. [latex]64 \cdot 4^{3x} = 16[/latex]
  2. [latex]2^{-3n} \cdot \dfrac{1}{4} = 2^{n+2}[/latex]
  3. [latex]\dfrac{36^{3x}}{36^{2x}} = 216^{2-b}[/latex]

For the following exercises, use logarithms to solve.

  1. [latex]9^{x-10} = 1[/latex]
  2. [latex]e^{r+10} - 10 = -42[/latex]
  3. [latex]-8 \cdot 10^{p+7} - 7 = -24[/latex]
  4. [latex]e^{-3k} + 6 = 44[/latex]
  5. [latex]-6e^{9x+8} + 2 = -74[/latex]
  6. [latex]e^{2x} - e^x - 132 = 0[/latex]

For the following exercise, use the definition of a logarithm to rewrite the equation as an exponential equation.

  1. [latex]\log\left(\dfrac{1}{100}\right) = -2[/latex]

For the following exercises, use the definition of a logarithm to solve the equation.

  1. [latex]5\log_7n = 10[/latex]
  2. [latex]4 + \log_2(9k) = 2[/latex]
  3. [latex]10 - 4\ln(9 - 8x) = 6[/latex]

For the following exercises, use the one-to-one property of logarithms to solve.

  1. [latex]\log_{13}(5n-2) = \log_{13}(8-5n)[/latex]
  2. [latex]\ln(-3x) = \ln(x^2-6x)[/latex]
  3. [latex]\ln(x-2) - \ln(x) = \ln(54)[/latex]
  4. [latex]\ln(x^2-10) + \ln(9) = \ln(10)[/latex]

For the following exercises, solve each equation for [latex]x[/latex].

  1. [latex]\ln(x) + \ln(x-3) = \ln(7x)[/latex]
  2. [latex]\ln(7) + \ln(2-4x^2) = \ln(14)[/latex]
  3. [latex]\ln(3) - \ln(3-3x) = \ln(4)[/latex]

Exponential and Logarithmic Models

For the following exercises, use the logistic growth model [latex]f(x)=\dfrac{150}{1+8e^{-2x}}[/latex].

  1. Find and interpret [latex]f(0)[/latex]. Round to the nearest tenth.
  2. Find the carrying capacity.
  3. Determine whether the data from the table could best be represented as a function that is linear, exponential, or logarithmic. Then write a formula for a model that represents the data.
    [latex]x[/latex] [latex]f(x)[/latex]
    [latex]-2[/latex] [latex]0.694[/latex]
    [latex]-1[/latex] [latex]0.833[/latex]
    [latex]0[/latex] [latex]1[/latex]
    [latex]1[/latex] [latex]1.2[/latex]
    [latex]2[/latex] [latex]1.44[/latex]
    [latex]3[/latex] [latex]1.728[/latex]
    [latex]4[/latex] [latex]2.074[/latex]
    [latex]5[/latex] [latex]2.488[/latex]

For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table could represent a function that is linear, exponential, or logarithmic.

  1. [latex]x[/latex] [latex]f(x)[/latex]
    [latex]1[/latex] [latex]2[/latex]
    [latex]2[/latex] [latex]4.079[/latex]
    [latex]3[/latex] [latex]5.296[/latex]
    [latex]4[/latex] [latex]6.159[/latex]
    [latex]5[/latex] [latex]6.828[/latex]
    [latex]6[/latex] [latex]7.375[/latex]
    [latex]7[/latex] [latex]7.838[/latex]
    [latex]8[/latex] [latex]8.238[/latex]
    [latex]9[/latex] [latex]8.592[/latex]
    [latex]10[/latex] [latex]8.908[/latex]
  2. [latex]x[/latex] [latex]f(x)[/latex]
    [latex]4[/latex] [latex]9.429[/latex]
    [latex]5[/latex] [latex]9.972[/latex]
    [latex]6[/latex] [latex]10.415[/latex]
    [latex]7[/latex] [latex]10.79[/latex]
    [latex]8[/latex] [latex]11.115[/latex]
    [latex]9[/latex] [latex]11.401[/latex]
    [latex]10[/latex] [latex]11.657[/latex]
    [latex]11[/latex] [latex]11.889[/latex]
    [latex]12[/latex] [latex]12.101[/latex]
    [latex]13[/latex] [latex]12.295[/latex]

For the following exercises, use a graphing calculator and this scenario: the population of a fish farm in [latex]t[/latex] years is modeled by the equation [latex]P(t)=\dfrac{1000}{1+9e^{-0.6t}}[/latex].

  1. Graph the function.
  2. To the nearest tenth, what is the doubling time for the fish population?
  3. To the nearest tenth, how long will it take for the population to reach [latex]900[/latex]?

For the following exercise, use this scenario: A doctor prescribes [latex]125[/latex] milligrams of a therapeutic drug that decays by about [latex]30%[/latex] each hour.

  1. Write an exponential model representing the amount of the drug remaining in the patient’s system after [latex]t[/latex] hours. Then use the formula to find the amount of the drug that would remain in the patient’s system after [latex]3[/latex] hours. Round to the nearest milligram.

For the following exercises, use this scenario: A tumor is injected with [latex]0.5[/latex] grams of Iodine-125, which has a decay rate of [latex]1.15%[/latex] per day.

  1. To the nearest day, how long will it take for half of the Iodine-125 to decay?
  2. A scientist begins with [latex]250[/latex] grams of a radioactive substance. After [latex]250[/latex] minutes, the sample has decayed to [latex]32[/latex] grams. Rounding to five decimal places, write an exponential equation representing this situation. To the nearest minute, what is the half-life of this substance?
  3. The half-life of Erbium-165 is [latex]10.4[/latex] hours. What is the hourly decay rate? Express the decimal result to four decimal places and the percentage to two decimal places.
  4. A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was [latex]1350[/latex] bacteria. Rounding to five decimal places, write an exponential equation representing this situation. To the nearest whole number, what is the population size after [latex]3[/latex] hours?

For the following exercise, use this scenario: A biologist recorded a count of [latex]360[/latex] bacteria present in a culture after [latex]5[/latex] minutes and [latex]1000[/latex] bacteria present after [latex]20[/latex] minutes.

  1. Rounding to six decimal places, write an exponential equation representing this situation. To the nearest minute, how long did it take the population to double?

For the following exercise, use this scenario: A pot of warm soup with an internal temperature of [latex]100^\circ[/latex] Fahrenheit was taken off the stove to cool in a [latex]69^\circ[/latex] F room. After fifteen minutes, the internal temperature of the soup was [latex]95^\circ[/latex] F.

  1. To the nearest minute, how long will it take the soup to cool to [latex]80^\circ[/latex] F?

For the following exercises, use this scenario: A turkey is taken out of the oven with an internal temperature of [latex]165^\circ[/latex] F and is allowed to cool in a [latex]75^\circ[/latex] F room. After half an hour, the internal temperature of the turkey is [latex]145^\circ[/latex] F.

  1. Write a formula that models this situation.
  2. To the nearest minute, how long will it take the turkey to cool to [latex]110^\circ[/latex] F?