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27.3: Review of exponential and logarithmic functions

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    Exercise \(\PageIndex{1}\)

    The population of a country grows exponentially at a rate of \(1\%\) per year. If the population was \(35.7\) million in the year \(2010\), then what is the population size of this country in the year \(2015\)?


    \(37.5\) million

    Exercise \(\PageIndex{2}\)

    A radioactive substance decays exponentially at a rate of \(7\%\) per hour. How long do you have to wait until the substance has decayed to \(\dfrac 1 4\) of its original size?


    \(19.1\) hours

    Exercise \(\PageIndex{3}\)

    Combine to an expression with only one logarithm.

    1. \(\dfrac{2}{3}\ln(x)+4\ln(y)\)
    2. \(\dfrac 1 2 \log_2(x)-\dfrac{3}{4}\log_2(y)+3\log_2(z)\)
    1. \(\ln \left(\sqrt[3]{x^{2}} y^{4}\right)\)
    2. \(\log _{2}\left(\dfrac{\sqrt{x} z^{3}}{\sqrt[4]{y^{3}}}\right)\)

    Exercise \(\PageIndex{4}\)

    Assuming that \(x,y>0\), write the following expressions in terms of \(u=\log(x)\) and \(v=\log(y)\):

    1. \(\log\left(\dfrac{\sqrt[3]{x^4}}{y^2}\right)\)
    2. \(\log\left(x\sqrt{y^5}\right)\)
    3. \(\log\left(\sqrt[5]{xy^4}\right)\)
    1. \(\dfrac{4}{3} u-2 v\)
    2. \(u+\dfrac{5}{2} v\)
    3. \(\dfrac{1}{5} u+\dfrac{4}{5} v\)

    Exercise \(\PageIndex{5}\)

    Solve without using the calculator: \(\log_3(x)+\log_3(x-8)=2\)



    Exercise \(\PageIndex{6}\)

    1. Find the exact solution of the equation: \(6^{x+2}=7^x\)
    2. Use the calculator to approximate your solution from part (a).
    1. \(x=\dfrac{2 \log 6}{\log 7-\log 6}\)
    2. \(x \approx 23.25\)

    Exercise \(\PageIndex{7}\)

    \(45\)mg of fluorine-18 decay in \(3\) hours to \(14.4\)mg. Find the half-life of fluorine-18.


    \(1.82\) hour

    Exercise \(\PageIndex{8}\)

    A bone has lost \(35\%\) of its carbon-\(14\). How old is the bone?


    \(3561\) year

    Exercise \(\PageIndex{9}\)

    How much do you have to invest today at \(3\%\) compounded quarterly to obtain \(\$ 2,000\) in return in \(3\) years?


    \(\$ 1,828.48\)

    Exercise \(\PageIndex{10}\)

    \(\$ 500\) is invested with continuous compounding. If \(\$692.01\) is returned after \(5\) years, what is the interest rate?


    \(r=6.5 \%\)

    This page titled 27.3: Review of exponential and logarithmic functions 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; a detailed edit history is available upon request.