Skip to main content
Mathematics LibreTexts

13.3: Exercises

  1. Last updated
  2. Save as PDF
  • Page ID
    49033

    • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

    ( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\id}{\mathrm{id}}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\kernel}{\mathrm{null}\,}\)

    \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\)

    \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\)

    \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    \( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

    \( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

    \( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vectorC}[1]{\textbf{#1}} \)

    \( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

    \( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

    \( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)

    Exercise \(\PageIndex{1}\)

    Graph the following functions with the calculator.

    1. \(y=5^x\)
    2. \(y=1.01^x\)
    3. \(y=\left (\dfrac 1 3 \right )^x\)
    4. \(y=0.97^x\)
    5. \(y=3^{-x}\)
    6. \(y=\left (\dfrac 1 3 \right )^{-x}\)
    7. \(y=e^{x^2}\)
    8. \(y=0.01^x\)
    9. \(y=1^x\)
    10. \(y=e^{x}+1\)
    11. \(y=\dfrac{e^x-e^{-x}}{2}\)
    12. \(y=\dfrac{e^x-e^{-x}}{e^x+e^{-x}}\)

    The last two functions are known as the hyperbolic sine, \(\sinh(x)=\dfrac{e^x-e^{-x}}{2}\), and the hyperbolic tangent, \(\tanh(x)=\dfrac{e^x-e^{-x}}{e^x+e^{-x}}\). Recall that the hyperbolic cosine \(\cosh(x)=\dfrac{e^x+e^{-x}}{2}\) was already graphed in example hyperbolic-cosine.

    Answer
    1. clipboard_e929886fc88b1e5ada4f15b02d6eec21a.png
    2. clipboard_eaf306e99bf7fb70aedacd464ecc7821a.png
    3. clipboard_eee8ff32d64022afbc5328894d6103f6c.png
    4. clipboard_eaf1f5a68c1e3100f5058af3da8458372.png
    5. same as c) since \(y=\left(\dfrac{1}{3}\right)^{x}=3^{-x}\)
    6. clipboard_ee2aa51fd0c9bf3df80bde31357a9ddde.png
    7. clipboard_e92023a7d7ae66b34e4a1bd81dd99fdd4.png
    8. clipboard_e8bf1d20fe2f629b9634fb50fc7352e2f.png
    9. clipboard_e9802b80547a2ceb8b2286b5a5ce7215d.png
    10. clipboard_e4cb72ca83788db3e3c3fa3027a69d5bd.png
    11. clipboard_e8a306ec0254118b085cb453931a64337.png
    12. clipboard_e6c76d1311825ca28327e82b37ea04265.png

    Exercise \(\PageIndex{2}\)

    Graph the given function. Describe how the graph is obtained by a transformation from the graph of an exponential function \(y=b^x\) (for appropriate base \(b\)).

    1. \(y=0.1\cdot 4^x\)
    2. \(y=3\cdot 2^x\)
    3. \(y=(-1)\cdot 2^x\)
    4. \(y=0.006\cdot 2^x\)
    5. \(y=e^{-x}\)
    6. \(y=e^{-x}+1\)
    7. \(y=(\dfrac 1 2)^{x}+3\)
    8. \(y=2^{x-4}\)
    9. \(y=2^{x+1}-6\)
    Answer
    1. \(y=4^{x}\) is compressed towards the x-axis by the factor \(0.1\) clipboard_e9a8a160a6cac2c21b6f0059ae20bcd1d.png
    2. \(y = 2^x\) stretched away from \(x\)-axis clipboard_eafcdca9fe269fe2e98bc970f88bfb13d.png
    3. \(y = 2^x\) reflected about the \(x\)-axis clipboard_e64e2e9be6cc3d7423cbe61a6b069fffb.png
    4. \(y = 2^x\) compressed towards the \(x\)-axis clipboard_ecb09bb72f8740e100c7b907d68685241.png
    5. \(y = e^x\) reflected about the \(y\)-axis clipboard_e1c93f931cc705f63520aebee0fa72207.png
    6. \(y = e^x\) reflected about the \(y\)-axis and shifted up by \(1\) clipboard_e11c91472da5fb45476dcba4a9321a715.png
    7. \(y=\left(\dfrac{1}{2}\right)^{x}\) shifted up by \(3\) clipboard_ec8ff260d6a742fe9db9b1667a93994eb.png
    8. \(y = 2^x\) shifted to the right by \(4\) clipboard_e442de036592464da648a4aa983f3c5fb.png
    9. \(y = 2^x\) shifted to the left by \(1\) and down by \(6\) clipboard_e530960e6d6e975674b92a314588f358a.png

    Exercise \(\PageIndex{3}\)

    Use the definition of the logarithm to write the given equation as an equivalent logarithmic equation.

    1. \(4^2=16\)
    2. \(2^{8}=256\)
    3. \(e^x=7\)
    4. \(10^{-1}=0.1\)
    5. \(3^x=12\)
    6. \(5^{7\cdot x}=12\)
    7. \(3^{2a+1}=44\)
    8. \(\left(\dfrac{1}{2}\right)^{\frac{x}{h}}=30\)
    Answer
    1. \(\log _{4}(16)=2\)
    2. \(\log _{2}(256)=8\)
    3. \(\ln (7)=x\)
    4. \(\log (0.1)=-1\)
    5. \(\log _{3}(12)=x\)
    6. \(\log _{5}(12)=7 x\)
    7. \(\log _{3}(44)=2 a+1\)
    8. \(\log _{\frac{1}{2}}(30)=\dfrac{x}{h}\)

    Exercise \(\PageIndex{4}\)

    Evaluate the following expressions without using a calculator.

    1. \(\log_7(49)\)
    2. \(\log_3(81)\)
    3. \(\log_{2}(64)\)
    4. \(\log_{50}(2,500)\)
    5. \(\log_2(0.25)\)
    6. \(\log(1,000)\)
    7. \(\ln(e^4)\)
    8. \(\log_{13}(13)\)
    9. \(\log(0.1)\)
    10. \(\log_6 \left (\dfrac 1 {36} \right)\)
    11. \(\ln(1)\)
    12. \(\log_{\frac 1 2}(8)\)
    Answer
    1. \(2\)
    2. \(4\)
    3. \(6\)
    4. \(2\)
    5. \(−2\)
    6. \(3\)
    7. \(4\)
    8. \(1\)
    9. \(−1\)
    10. \(−2\)
    11. \(0\)
    12. \(−3\)

    Exercise \(\PageIndex{5}\)

    Using a calculator, approximate the following expressions to the nearest thousandth.

    1. \(\log_3(50)\)
    2. \(\log_3(12)\)
    3. \(\log_{17}(0.44)\)
    4. \(\log_{0.34}(200)\)
    Answer
    1. \(3.561\)
    2. \(2.262\)
    3. \(−0.290\)
    4. \(−4.911\)

    Exercise \(\PageIndex{6}\)

    State the domain of the function \(f\) and sketch its graph.

    1. \(f(x)=\log(x)\)
    2. \(f(x)=\log(x+7)\)
    3. \(f(x)=\ln(x+5)-1\)
    4. \(f(x)=\ln(3x-6)\)
    5. \(f(x)=2\cdot \log(x+4)\)
    6. \(f(x)=-4\cdot\log(x+2)\)
    7. \(f(x)=\log_{3}(x-5)\)
    8. \(f(x)=-\ln(x-1)+3\)
    9. \(f(x)=\log_{0.4}(x)\)
    10. \(f(x)=\log_{3}(-5x)-2\)
    11. \(f(x)=\log|x|\)
    12. \(f(x)=\log|x+2|\)
    Answer
    1. clipboard_e5160d6465097203d350bcd2c73ba1d79.png
    2. clipboard_e36290957ff8262fdc2f5b06123dd0450.png
    3. clipboard_e45c9904d3ef9aab9b868baa6dccf3ff9.png
    4. clipboard_e7f25ca69dc3b229c82c273c2d225a03f.png
    5. clipboard_e93556a07118703506dd7b90588402b0f.png
    6. clipboard_e15bd3ae096926c15de10492ea301dcdb.png
    7. clipboard_e881f85469a7a85da9f74c62cb8830c68.png
    8. clipboard_ea5de6be00b3734051c6ed121939bb153.png
    9. clipboard_eda23b0f1b489994463fbe02ea30be943.png
    10. clipboard_e379f1647d1818fd49c2353387dc0cd6e.png
    11. clipboard_ef7d967f62b89020db412de5ad48ec63e.png
    12. clipboard_ecd2053a9dae8cc7f605c9ce224d4cf02.png

    This page titled 13.3: Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Thomas Tradler and Holly Carley (New York City College of Technology at CUNY Academic Works) via source content that was edited to the style and standards of the LibreTexts platform.