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Mathematics LibreTexts

4.R: Exponential and Logarithmic Functions (Review)

  • Page ID
    18199
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    4.1: Exponential Functions

    1) Determine whether the function \(y=156(0.825)^t\) represents exponential growth, exponential decay, or neither. Explain.

    Answer

    exponential decay; The growth factor, \(0.825\), is between \(0\) and \(1\).

    2) The population of a herd of deer is represented by the function \(A(t)=205(1.13)^t\),where \(t\) is given in years. To the nearest whole number, what will the herd population be after \(6\) years?

    3) Find an exponential equation that passes through the points \((2,2.25)\) and \((5,60.75)\).

    Answer

    4) Determine whether Table below could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points.

    x 1 2 3 4
    f(x) 3 0.9 0.27 0.081

    5) A retirement account is opened with an initial deposit of \(\$8,500\) and earns \(8.12\%\)interest compounded monthly. What will the account be worth in \(20\) years?

    Answer

    6) Hsu-Mei wants to save \(\$5,000\) for a down payment on a car. To the nearest dollar, how much will she need to invest in an account now with \(7.5\%\) APR, compounded daily, in order to reach her goal in \(3\) years?

    7) Does the equation \(y=2.294e^{-0.654t}\) represent continuous growth, continuous decay, or neither? Explain.

    Answer

    continuous decay; the growth rate is negative.

    8) Suppose an investment account is opened with an initial deposit of \(\$10,500\) earning \(6.25\%\) interest, compounded continuously. How much will the account be worth after \(25\) years?

    4.2: Graphs of Exponential Functions

    1) Graph the function \(f(x)=3.5(2)^x\).State the domain and range and give the \(y\)-intercept.

    Answer

    domain: all real numbers; range: all real numbers strictly greater than zero; \(y\)-intercept: \((0, 3.5)\);

    CNX_PreCalc_Figure_04_08_229.jpg

    2) Graph the function \(f(x)=4\left(\dfrac{1}{8}\right)^x\) and its reflection about the \(y\)-axis on the same axes, and give the \(y\)-intercept.

    3) The graph of \(f(x)=6.5^x\) is reflected about the \(y\)-axis and stretched vertically by a factor of \(7\).What is the equation of the new function, \(g(x)\)? State its \(y\)-intercept, domain, and range.

    Answer

    4) The graph below shows transformations of the graph of \(f(x)=2^x\).What is the equation for the transformation?

    CNX_PreCalc_Figure_04_08_231.jpg

    4.3: Logarithmic Functions

    1) Rewrite \(\log_{17}(4913)=x\) as an equivalent exponential equation.

    Answer

    \(17^x=4913\)

    2) Rewrite \(\ln(s)=t\) as an equivalent exponential equation.

    3) Rewrite \(a^{-\tfrac{2}{5}}=b\) as an equivalent logarithmic equation.

    Answer

    \(\log_a b=-\dfrac{2}{5}\)

    4) Rewrite \(e^{-3.5}=h\) as an equivalent logarithmic equation.

    5) Solve for \(x\) if \(\log_{64}(x)=\dfrac{1}{3}\) by converting to exponential form.

    Answer

    \(x=64^{\tfrac{1}{3}}=4\)

    6) Evaluate \(\log_5\left(\dfrac{1}{125}\right)\) without using a calculator.

    7) Evaluate \(\log(0.000001)\) without using a calculator.

    Answer

    \(\log(0.000001)=-6\)

    8) Evaluate \(\log(4.005)\) using a calculator. Round to the nearest thousandth.

    9) Evaluate \(\ln\left(e^{-0.8648}\right)\) without using a calculator.

    Answer

    \(\ln\left(e^{-0.8648}\right)=-0.8648\)

    10) Evaluate \(\ln \left ( \sqrt[3]{18} \right )\) using a calculator. Round to the nearest thousandth.

    4.4: Graphs of Logarithmic Functions

    1) Graph the function \(g(x)=\log(7x+21)-4\)

    Answer

    ExR 4.4.1.png

    2) Graph the function \(h(x)=2\ln(9-3x)+1\)

    3) State the domain, vertical asymptote, and end behavior of the function \(g(x)=\ln(4x+20)-17\).

    Answer

    Domain: \(x>-5\);Vertical asymptote: \(x=-5\);End behavior: \(x\rightarrow -5^+,f(x)\rightarrow -\infty\) and as \(x\rightarrow \infty ,f(x)\rightarrow \infty\).

    4.5: Logarithmic Properties

    1) Rewrite \(\ln(7r\cdot 11st)\) in expanded form.

    2) Rewrite \(\log_8(x)+\log_8(5)+\log_8(y)+\log_8(13)\) in compact form.

    Answer

    \(\log_8(65xy)\)

    3) Rewrite \(\log_m\left(\dfrac{67}{83}\right)\) in expanded form.

    4) Rewrite \(\ln(z)-\ln(x)-\ln(y)\) in compact form.

    Answer

    \(\ln \left(\dfrac{z}{xy}\right)\)

    5) Rewrite \(\ln \left(\dfrac{1}{x^5}\right)\) as a product.

    6) Rewrite \(-\log_y \left(\dfrac{1}{12}\right)\) as a single logarithm.

    Answer

    \(\log_y (12)\)

    7) Use properties of logarithms to expand \(\log \left(\dfrac{r^2s^{11}}{t^{14}}\right)\).

    8) Use properties of logarithms to expand \(\ln \left(2b\sqrt{\dfrac{b+1}{b-1}}\right)\)

    Answer

    \(\ln(2)+\ln(b)+\dfrac{\ln(b+1)-\ln(b-1)}{2}\)

    9) Condense the expression \(5\ln(b)+\ln(c)+\dfrac{\ln(4-a)}{2}\) to a single logarithm.

    10) Condense the expression \(3\log_7 v+6\log_7 w-\dfrac{\log_7 u}{3}\) to a single logarithm.

    Answer

    \(\log_7 \left (\dfrac{v^3 w^6}{\sqrt[3]{u}} \right )\)

    11) Rewrite \(\log_3(12.75)\) to base

    12) Rewrite \(5^{12x-17}=125\) as a logarithm. Then apply the change of base formula to solve for \(x\) using the common log. Round to the nearest thousandth.

    Answer

    \(x = \dfrac{\tfrac{\log (125)}{\log (5)}+17}{12}=\dfrac{5}{3}\)

    4.6: Exponential and Logarithmic Equations

    1) Solve \(216^{3x}\cdot 216^x=36^{3x+2}\) by rewriting each side with a common base.

    2) Solve \(\dfrac{125}{\left(\tfrac{1}{625}\right)^{-x-3}}=5^3\) by rewriting each side with a common base.

    Answer

    \(x=-3\)

    3) Use logarithms to find the exact solution for \(7\cdot 17^{-9x}-7=49\). If there is no solution, write no solution.

    4) Use logarithms to find the exact solution for \(3e^{6n-2}+1=-60\). If there is no solution, write no solution.

    Answer

    no solution

    5) Find the exact solution for \(5e^{3x}-4=6\). If there is no solution, write no solution.

    6) Find the exact solution for \(2e^{5x-2}-9=-56\). If there is no solution, write no solution.

    Answer

    no solution

    7) Find the exact solution for \(5^{2x-3}=7^{x+1}\). If there is no solution, write no solution.

    8) Find the exact solution for \(e^{2x}-e^x-110=0\). If there is no solution, write no solution.

    Answer

    \(x=\ln (11)\)

    9) Use the definition of a logarithm to solve: \(-5\log_7(10n)=5\)

    10) Use the definition of a logarithm to find the exact solution for \(9+6\ln(a+3)=33\).

    Answer

    \(a=e^4-3\)

    11) Use the one-to-one property of logarithms to find an exact solution for \(\log_8(7)+\log_8(-4x)=\log_8(5)\). If there is no solution, write no solution.

    12) Use the one-to-one property of logarithms to find an exact solution for \(\ln(5)+\ln(5x^2-5)=\ln(56)\). If there is no solution, write no solution.

    Answer

    \(x=\pm \dfrac{9}{5}\)

    13) The formula for measuring sound intensity in decibels \(D\) is defined by the equation \(D=10\log \left(\dfrac{I}{I_0}\right)\), where \(I\) is the intensity of the sound in watts per square meter and \(I_0=10^{-12}\) is the lowest level of sound that the average person can hear. How many decibels are emitted from a large orchestra with a sound intensity of \(6.3\cdot 10^{-3}\) watts per square meter?

    14) The population of a city is modeled by the equation \(P(t)=256,114e^{0.25t}\) where \(t\) is measured in years. If the city continues to grow at this rate, how many years will it take for the population to reach one million?

    Answer

    about \(5.45\) years

    15) Find the inverse function \(f^{-1}\) for the exponential function \(f(x)=2\cdot e^{x+1}-5\).

    16) Find the inverse function \(f^{-1}\) for the logarithmic function \(f(x)=0.25\cdot \log_2(x^3+1)\)

    Answer

    \(f^{-1}(x)=\sqrt[3]{2^{4x}-1}\)

    4.7: Exponential and Logarithmic Models

    For the exercises 1-2, use this scenario: A doctor prescribes \(300\) milligrams of a therapeutic drug that decays by about \(17\%\) each hour.

    1) To the nearest minute, what is the half-life of the drug?

    2) Write an exponential model representing the amount of the drug remaining in the patient’s system after \(t\) hours. Then use the formula to find the amount of the drug that would remain in the patient’s system after \(24\) hours. Round to the nearest hundredth of a gram.

    Answer

    \(f(t)=300(0.83)^t;f(24)\approx 3.43 \text{ g}\)

    For the exercises 3-4, use this scenario: A soup with an internal temperature of \(350^{\circ}\) F was taken off the stove to cool in a \(71^{\circ}\) F room. After fifteen minutes, the internal temperature of the soup was \(175^{\circ}\) F.

    3) Use Newton’s Law of Cooling to write a formula that models this situation.

    4) How many minutes will it take the soup to cool to \(85^{\circ}\)?

    Answer

    about \(45\) minutes

    For the exercises 5-7, use this scenario: The equation \(N(t)=\dfrac{1200}{1+199e^{-0.625t}}\) models the number of people in a school who have heard a rumor after \(t\) days.

    5) How many people started the rumor?

    6) To the nearest tenth, how many days will it be before the rumor spreads to half the carrying capacity?

    Answer

    about \(8.5\) days

    7) What is the carrying capacity?

    For the exercises 8-10, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table would likely represent a function that is linear, exponential, or logarithmic.

    8)

    \(x\) \(f(x)\)
    1 3.05
    2 4.42
    3 6.4
    4 9.28
    5 13.46
    6 19.52
    7 28.3
    8 41.04
    9 59.5
    10 86.28
    Answer

    exponential

    ExR 4.7.8.png

    9)

    \(x\) \(f(x)\)
    0.5 18.05
    1 17
    3 15.33
    5 14.55
    7 14.04
    10 13.5
    12 13.22
    13 13.1
    15 12.88
    17 12.69
    20 12.45

    10) Find a formula for an exponential equation that goes through the points \((-2,100)\) and \((0,4)\).Then express the formula as an equivalent equation with base \(e\).

    Answer

    \(y=4(0.2)^x; y=4e^{-1.609438x}\)

    4.8: Fitting Exponential Models to Data

    1) What is the carrying capacity for a population modeled by the logistic equation \(P(t)=\dfrac{250,000}{1+499e^{-0.45t}}\)? What is the initial population for the model?

    2) The population of a culture of bacteria is modeled by the logistic equation \(P(t)=\dfrac{14,250}{1+29e^{-0.62t}}\), where \(t\) is in days. To the nearest tenth, how many days will it take the culture to reach \(75\%\) of its carrying capacity?

    Answer

    about \(7.2\) days

    For the exercises 3-5 use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places.

    3)

    \(x\) \(f(x)\)
    1 409.4
    2 260.7
    3 170.4
    4 110.6
    5 74
    6 44.7
    7 32.4
    8 19.5
    9 12.7
    10 8.1

    4)

    \(x\) \(f(x)\)
    0.15 36.21
    0.25 28.88
    0.5 24.39
    0.75 18.28
    1 16.5
    1.5 12.99
    2 9.91
    2.25 8.57
    2.75 7.23
    3 5.99
    3.5 4.81
    Answer

    logarithmic; \(y=16.68718-9.71860\ln(x)\)

    ExR 4.8.4.png

    5)

    \(x\) \(f(x)\)
    0 9
    2 22.6
    4 44.2
    5 62.1
    7 96.9
    8 113.4
    10 133.4
    11 137.6
    15 148.4
    17 149.3

    Practice Test

    1) The population of a pod of bottlenose dolphins is modeled by the function \(A(t)=8(1.17)^t\), where \(t\) is given in years. To the nearest whole number, what will the pod population be after \(3\) years?

    Answer

    About \(13\) dolphins.

    2) Find an exponential equation that passes through the points \((0,4)\) and \((2,9)\).

    3) Drew wants to save \(\$2,500\) to go to the next World Cup. To the nearest dollar, how much will he need to invest in an account now with \(6.25\%\) APR, compounding daily, in order to reach his goal in \(4\) years?

    Answer

    \(\$1,947\)

    4) An investment account was opened with an initial deposit of \(\$9,600\)and earns \(7.4\%\) interest, compounded continuously. How much will the account be worth after \(15\) years?

    5) Graph the function \(f(x)=5(0.5)^{-x}\) and its reflection across the \(y\)-axis on the same axes, and give the \(y\)-intercept.

    Answer

    \(y\)-intercept:

    ExP 4.P.5.png

    6) The graph shows transformations of the graph of \(f(x)=\left(\dfrac{1}{2}\right)^x\).What is the equation for the transformation?

    Ex4RP.png

    7) Rewrite \(\log_{8.5}(614.125)=a\) as an equivalent exponential equation.

    Answer

    \(8.5^a=614.125\)

    8) Rewrite \(e^{\tfrac{1}{2}}=m\) as an equivalent logarithmic equation.

    9) Solve for \(x\) by converting the logarithmic equation \(\log_{\tfrac{1}{7}}(x)=2\) to exponential form.

    Answer

    \(x=\left(\dfrac{1}{7}\right)^2=\dfrac{1}{49}\)

    10) Evaluate \(\log(10,000,000)\) without using a calculator.

    11) Evaluate \(\ln(0.716)\) using a calculator. Round to the nearest thousandth.

    Answer

    \(\ln(0.716)\approx -0.334\)

    12) Graph the function

    13) State the domain, vertical asymptote, and end behavior of the function

    Answer

    Domain: \(x<3\);Vertical asymptote: \(x=3\);End behavior: \(x\rightarrow 3^-,f(x)\rightarrow -\infty\) and \(x\rightarrow -\infty ,f(x)\rightarrow \infty\)

    14) Rewrite \(\log(17a\cdot 2b)\) as a sum.

    15) Rewrite \(\log_t(96)-\log_t(8)\) in compact form.

    Answer

    \(\log_t(12)\)

    16) Rewrite \(\log_8 \left(a^{\tfrac{1}{b}}\right)\) as a product.

    17) Use properties of logarithm to expand \(\ln(y^3z^2\cdot \sqrt[3]{x-4})\).

    Answer

    \(3\ln(y)+2\ln (z)+\dfrac{(x-4)}{3}\)

    18) Condense the expression \(4\ln(c)+\ln (d)+\dfrac{\ln}{3}\dfrac{(b+3)}{3}\) to a single logarithm.

    19) Rewrite \(16^{3x-5}=1000\) as a logarithm. Then apply the change of base formula to solve for \(x\) using the natural log. Round to the nearest thousandth.

    Answer

    \(x = \dfrac{\tfrac{\ln(1000)}{\ln(16)}+5}{3}\approx 2.497\)

    20) Solve \(\left ( \dfrac{1}{81} \right )^x\cdot \dfrac{1}{243}=\left ( \dfrac{1}{9} \right )^{-3x-1}\) by rewriting each side with a common base.

    21) Use logarithms to find the exact solution for \(-9e^{10a-8}-5=-41\). If there is no solution, write no solution.

    Answer

    \(a=\dfrac{\ln(4)+8}{10}\)

    22) Find the exact solution for \(10e^{4x+2}+5=56\).If there is no solution, write no solution.

    23) Find the exact solution for \(-5e^{-4x-1}-4=64\).If there is no solution, write no solution.

    Answer

    no solution

    24) Find the exact solution for \(2^{x-3}=6^{2x-1}\).If there is no solution, write no solution.

    25) Find the exact solution for \(e^{2x}-e^x-72=0\).If there is no solution, write no solution.

    Answer

    \(x=\ln(9)\)

    26) Use the definition of a logarithm to find the exact solution for \(4\log(2n)-7=-11\).

    27) Use the one-to-one property of logarithms to find an exact solution for \(\log(4x^2-10)+\log(3)=\log(51)\). If there is no solution, write no solution.

    Answer

    \(x=\pm \dfrac{3\sqrt{3}}{2}\)

    28) The formula for measuring sound intensity in decibels \(D\) is defined by the equation \(D=10\log\left ( \dfrac{I}{I_0} \right )\),where \(I\) is the intensity of the sound in watts per square meter and \(I_0=10^{-12}\)is the lowest level of sound that the average person can hear. How many decibels are emitted from a rock concert with a sound intensity of \(4.7\cdot 10^{-1}\) watts per square meter?

    29) A radiation safety officer is working with \(112\) grams of a radioactive substance. After \(17\) days, the sample has decayed to \(80\) grams. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest day, what is the half-life of this substance?

    Answer

    \(f(t)=112e^{-.019792t}\); half-life: about \(35\) days

    30) Write the formula found in the previous exercise as an equivalent equation with base \(e\).Express the exponent to five significant digits.

    31) A bottle of soda with a temperature of \(71^{\circ}\)Fahrenheit was taken off a shelf and placed in a refrigerator with an internal temperature of \(35^{\circ}\) F.After ten minutes, the internal temperature of the soda was \(63^{\circ}\) F.Use Newton’s Law of Cooling to write a formula that models this situation. To the nearest degree, what will the temperature of the soda be after one hour?

    Answer

    \(T(t)=36e^{-0.025131t}+35\); \(T(60)\approx 43^{\circ}\) F

    32) Enter the data from Table into a graphing calculator and graph the resulting scatter plot. Determine whether the data from the table would likely represent a function that is linear, exponential, or logarithmic.

    \(x\) \(f(x)\)
    1 3
    2 8.55
    3 11.79
    4 14.09
    5 15.88
    6 17.33
    7 18.57
    8 19.64
    9 20.58
    10 21.42
    Answer

    logarithmic

    4RP1.png

    33) The population of a lake of fish is modeled by the logistic equation \(P(t)=\dfrac{16,120}{1+25e^{-0.75t}}\), where \(t\) is time in years. To the nearest hundredth, how many years will it take the lake to reach \(80\%\) of its carrying capacity?

    For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places.

    34)

    \(x\) \(f(x)\)
    1 20
    2 21.6
    3 29.2
    4 36.4
    5 46.6
    6 55.7
    7 72.6
    8 87.1
    9 107.2
    10 138.1
    Answer

    exponential; \(y=15.10062(1.24621)^x\)

    4RP34.png

    35)

    \(x\) \(f(x)\)
    3 13.98
    4 17.84
    5 20.01
    6 22.7
    7 24.1
    8 26.15
    9 27.37
    10 28.38
    11 29.97
    12 31.07
    13 31.43

    36)

    \(x\) \(f(x)\)
    0 2.2
    0.5 2.9
    1 3.9
    1.5 4.8
    2 6.4
    3 9.3
    4 12.3
    5 15
    6 16.2
    7 17.3
    8 17.9
    Answer

    logistic; \(y=\dfrac{18.41659}{1+7.54644e^{-0.68375x}}\)

    4RP36.png

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