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Mathematics LibreTexts

4.5E: Exercises

  • Page ID
    18585
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    4.5: Logarithmic Properties

    Verbal

    1) How does the power rule for logarithms help when solving logarithms with the form \(\log _b(\sqrt[n]{x})\)?

    Answer

    Any root expression can be rewritten as an expression with a rational exponent so that the power rule can be applied, making the logarithm easier to calculate. Thus, \(\log _b \left ( x^{\frac{1}{n}} \right ) = \dfrac{1}{n}\log_{b}(x)\).

    2) What does the change-of-base formula do? Why is it useful when using a calculator?

    Algebraic

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

    3) \(\log _b (7x\cdot 2y)\)

    Answer

    \(\log _b (2)+\log _b (7)+\log _b (x)+\log _b (y)\)

    4) \(\ln (3ab\cdot 5c)\)

    5) \(\log_b \left ( \dfrac{13}{17} \right )\)

    Answer

    \(\log _b (13)-\log _b (17)\)

    6) \(\log_4 \left ( \dfrac{\frac{x}{z}}{w} \right )\)

    7) \(\ln \left ( \dfrac{1}{4^k} \right )\)

    Answer

    \(-k\ln(4)\)

    8) \(\log _2 (y^x)\)

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

    9) \(\ln (7)+\ln (x)+\ln (y)\)

    Answer

    \(\ln(7xy)\)

    10) \(\log_3(2)+\log_3(a)+\log_3(11)+\log_3(b)\)

    11) \(\log_b(28)-\log_b(7)\)

    Answer

    \(\log_b(4)\)

    12) \(\ln (a)-\ln (d)-\ln (c)\)

    13) \(-\log_b\left ( \dfrac{1}{7} \right )\)

    Answer

    \(\log_b(7)\)

    14) \(\dfrac{1}{3}\ln(8)\)

    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.

    15) \(\log \left ( \dfrac{x^{15}y^{13}}{z^{19}} \right )\)

    Answer

    \(15\log (x)+13\log (y)-19\log (z)\)

    16) \(\ln \left ( \frac{a^{-2}}{b^{-4}c^{5}} \right )\)

    17) \(\log \left ( \sqrt{x^3y^{-4}} \right )\)

    Answer

    \(\frac{3}{2}\log (x)-2\log (y)\)

    18) \(\ln \left ( y\sqrt{\frac{y}{1-y}} \right )\)

    19) \(\log \left ( x^2y^3 \sqrt[3]{x^2y^5} \right )\)

    Answer

    \(\dfrac{8}{3}\log (x)+\dfrac{14}{3}\log (y)\)

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

    20) \(\log \left ( 2x^4 \right )+\log \left (3x^5 \right )\)

    21) \(\ln \left ( 6x^9 \right )-\ln \left (3x^2 \right )\)

    Answer

    \(\ln \left ( 2x^7 \right )\)

    22) \(2\log (x)+3\log (x+1)\)

    23) \(\log (x)-\dfrac{1}{2}\log (y)+3\log (z)\)

    Answer

    \(\log \left ( \dfrac{xz^3}{\sqrt{y}} \right )\)

    24) \(4\log _7(c)+\dfrac{\log _7(a)}{3}+\dfrac{\log _7(b)}{3}\)

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

    25) \(\log _7(15)\) to base \(e\)

    Answer

    \(\log _7(15)=\dfrac{\ln (15)}{\ln (7)}\)

    26) \(\log _{14}(55.875)\) to base \(10\) 

    For the following exercises, suppose \(\log _5(6)=a\) and \(\log _5(11)=b\) .Use the change-of-base formula along with properties of logarithms to rewrite each expression in terms of \(a\) and \(b\) .Show the steps for solving.  

    27) \(\log _{11} (5)\)

    Answer

    \(\log _{11} (5)=\dfrac{\log_5 (5)}{\log_5 (11)}=\dfrac{1}{b}\)

    28) \(\log _{6} (55)\)

    29) \(\log _{11}\left (\dfrac{6}{11} \right )\)

    Answer

    \(\log _{11}\left (\dfrac{6}{11} \right )=\dfrac{\log _{11}\left (\frac{6}{11} \right )}{\log _{5}(11)}=\dfrac{\log _{5}(6)-\log _{5}(11)}{\log _{5}(11)}=\dfrac{a-b}{b}=\dfrac{a}{b}-1\)

    Numeric

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

    30) \(\log _3 \left ( \dfrac{1}{9} \right )-3\log _3 (3)\)

    31) \(6\log _8 (2)+\dfrac{\log _8 (64)}{3\log _8 (4)}\)

    Answer

    \(3\)

    32) \(2\log _9 (3)-4\log _9 (3)+\log _9 \left (\dfrac{1}{729} \right )\)

    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.

    33) \(\log _3 (22)\)

    Answer

    \(2.81359\)

    34) \(\log _8 (65)\)

    35) \(\log _6 (5.38)\)

    Answer

    \(0.93913\)

    36) \(\log _4 \left (\dfrac{15}{2} \right )\)

    37) \(\log _{\frac{1}{2}} (4.7)\)

    Answer

    \(-2.23266\)

    Extensions

    38) Use the product rule for logarithms to find all \(x\) values such that \(\log _{12} (2x+6)+\log _{12} (x+2)=2\).Show the steps for solving.

    39) Use the quotient rule for logarithms to find all \(x\) values such that \(\log _{6} (x+2)-\log _{6} (x-3)=1\).Show the steps for solving.

    Answer

    Rewriting as an exponential equation and solving for \(x\):

    \(\begin{align*}
    6^1 &= \frac{x+2}{x-3}\\
    0 &= \frac{x+2}{x-3}-6\\
    0 &= \frac{x+2}{x-3}-\frac{6(x-3)}{(x-3)}\\
    0 &= \frac{x+2-6x+18}{x-3}\\
    0 &= \frac{x-4}{x-3}\\
    x &= 4
    \end{align*}\)

    Checking, we find that \(\log _6(4+2)-\log _6(4-3)=\log _6(6)-\log _6(1)\) is defined, so \(x=4\)

    40) Can the power property of logarithms be derived from the power property of exponents using the equation \(b^x=m\)?If not, explain why. If so, show the derivation.

    41) Prove that \(\log_b(n)=\frac{1}{\log_b(n)}\) for any positive integers \(b>1\) and \(n>1\).

    Answer

    Let \(b\) and \(n\) be positive integers greater than \(1\).Then, by the change-of-base formula, \(\log_b(n)=\frac{\log_n(n)}{\log_n(b)}=\frac{1}{\log_n(b)}\)

    42) Does \(\log_{81}(2401)=\log_3(7)\)?Verify the claim algebraically.