Exercises - Properties of Logarithms

  1. Find the values of the following:

    1. $\displaystyle{\log_4 8}$  

    2. $\displaystyle{\log_{1/9} \sqrt{27}}$  

    3. $\displaystyle{\ln \sqrt[3]{e^2}}$  

  2. Solve for the unknown:

    1. $\displaystyle{\log_b 16 = \frac{4}{3}}$  

    2. $\displaystyle{\ln x = -1}$  

  3. Simplify:

    1. $\displaystyle{2^{\log_2 5x}}$  

    2. $\displaystyle{\ln \left( \frac{e^5}{e^3} \right)}$  

    3. $\displaystyle{\log_5 (3-2)}$  

  4. Write as a single logarithm:

    1. $\displaystyle{\ln 3 + 2 \ln 4}$  

    2. $\displaystyle{\frac{1}{2} \log_5 49 - \frac{1}{3} \log_5 8 + 13 \log_5 1}$  

  5. Simplify $\displaystyle{e^{2\ln 3 - 3\ln 2}}$  

  6. Use the approximate values $\displaystyle{\log_{10} 2 \approx 0.3010 \, \textrm{and} \, \log_{10} 3 \approx 0.4771}$ to compute

    1. $\displaystyle{\log_{10} 24}$

    2. $\displaystyle{\log_{10} 5}$

    3. $\displaystyle{\log_{10} \sqrt[3]{4}}$

  7. Rewrite the given exponential statement in logarithmic form:

    1. $\displaystyle{4^{-1/2} = \frac{1}{2}}$   $\ans{\displaystyle{\log_4 \frac{1}{2} = -\frac{1}{2}}}$

    2. $\displaystyle{9^0 = 1}$   $\ans{\displaystyle{\log_9 1 = 0}}$

    3. $\displaystyle{10^y = x}$   $\ans{\displaystyle{\log_{10} x = y}}$

    4. $\displaystyle{e^y = 3}$   $\ans{\displaystyle{\ln 3 = y}}$

    5. $\displaystyle{\left( \frac{1}{64} \right)^{-1/2} = 8}$   $\ans{\displaystyle{\log_{\frac{1}{64}} 8 = - \frac{1}{2}}}$

    6. $\displaystyle{e^5 = x}$   $\ans{\displaystyle{\ln x = 5}}$

    7. $\displaystyle{36^{-3/2} = \frac{1}{216}}$   $\ans{\displaystyle{\log_{36} \frac{1}{216} = -\frac{3}{2}}}$

  8. Rewrite the given logarithmic statement in exponential form:

    1. $\displaystyle{\log_3 81 = 4}$   $\ans{\displaystyle{3^4 = 81}}$

    2. $\displaystyle{\log_{10} 10 = 1}$   $\ans{\displaystyle{10^{1} = 10}}$

    3. $\displaystyle{\log_x 5 = 2}$   $\ans{\displaystyle{x^2 = 5}}$

    4. $\displaystyle{\log_2 x = y}$   $\ans{\displaystyle{2^y = x}}$

    5. $\displaystyle{\ln x = 3}$   $\ans{\displaystyle{e^3 = x}}$

    6. $\displaystyle{\log_5 \frac{1}{25} = -2}$   $\ans{\displaystyle{5^{-2} = \frac{1}{25}}}$

    7. $\displaystyle{\ln 3x = -2}$   $\ans{\displaystyle{e^{-2} = 3x}}$

    8. $\displaystyle{\log_{16} 2 = \frac{1}{4}}$   $\ans{\displaystyle{16^{\frac{1}{4}} = 2}}$

    9. $\displaystyle{\ln e^2 = 2}$   $\ans{\displaystyle{e^2 = e^2}}$

  9. Find the value of the following:

    1. $\displaystyle{\log_{10} 0.000001}$   $\ans{\displaystyle{-6}}$

    2. $\displaystyle{\log_2 (2^2+2^2)}$   $\ans{\displaystyle{3}}$

    3. $\displaystyle{\log_{64} \frac{1}{32}}$   $\ans{\displaystyle{-\frac{5}{6}}}$

    4. $\displaystyle{\log_{1/2} 16}$   $\ans{\displaystyle{-4}}$

    5. $\displaystyle{\ln \sqrt{e}}$   $\ans{\displaystyle{\frac{1}{2}}}$

    6. $\displaystyle{\log_{5/2} \frac{8}{125}}$   $\ans{\displaystyle{-3}}$

    7. $\displaystyle{\ln (e^2 \cdot e^3)}$   $\ans{\displaystyle{5}}$

    8. $\displaystyle{\ln (e^2)^3}$   $\ans{\displaystyle{6}}$

    9. $\displaystyle{\log_4 \frac{1}{64}}$   $\ans{\displaystyle{-3}}$

    10. $\displaystyle{\log_7 \sqrt[3]{49}}$   $\ans{\displaystyle{\frac{2}{3}}}$

    11. $\displaystyle{\log_{\sqrt{3}} 9}$   $\ans{\displaystyle{4}}$

    12. $\displaystyle{\log_8 \frac{1}{4}}$   $\ans{\displaystyle{-\frac{2}{3}}}$

    13. $\displaystyle{\log_6 216}$   $\ans{\displaystyle{3}}$

    14. $\displaystyle{\ln \frac{1}{\sqrt[3]{e^2}}}$   $\ans{\displaystyle{-\frac{2}{3}}}$

    15. $\displaystyle{\ln \, \left( \frac{e^{3/2}}{e^2 \sqrt{e}} \right)}$   $\ans{\displaystyle{-1}}$

  10. Solve for the unknown.

    1. $\displaystyle{\log_b 125 = 3}$   $\ans{\displaystyle{5}}$

    2. $\displaystyle{\log_7 343 = x}$   $\ans{\displaystyle{3}}$

    3. $\displaystyle{\log_2 \frac{1}{N} = 5}$   $\ans{\displaystyle{\frac{1}{32}}}$

    4. $\displaystyle{2\log_9 x = 1}$   $\ans{\displaystyle{3}}$

    5. $\displaystyle{\log_3 \frac{1}{27} = x}$   $\ans{\displaystyle{-3}}$

    6. $\displaystyle{\ln x = 3}$   $\ans{\displaystyle{e^3}}$

    7. $\displaystyle{\ln \sqrt{e} = x}$   $\ans{\displaystyle{\frac{1}{2}}}$

    8. $\displaystyle{\log_{10} N = -2}$   $\ans{\displaystyle{0.01}}$

    9. $\displaystyle{\log_5 25^c = 4}$   $\ans{\displaystyle{2}}$

    10. $\displaystyle{\ln e^{2x} = -\frac{1}{2}}$   $\ans{\displaystyle{-\frac{1}{4}}}$

    11. $\displaystyle{\log_x 6 = -1}$   $\ans{\displaystyle{\frac{1}{6}}}$

    12. $\displaystyle{\log_2 4^{-3} = x}$   $\ans{\displaystyle{-6}}$

    13. $\displaystyle{\ln e^4 = x^2}$   $\ans{\displaystyle{2,-2}}$

    14. $\displaystyle{\log_{10} \, \left( \frac{1}{1000} \right)^x = 1}$   $\ans{\displaystyle{-\frac{1}{3}}}$

  11. Use the approximate values $\log_{10} 4 \doteq 0.6021$ and $\log_{10} 5 \doteq 0.6990$ to compute:

    1. $\displaystyle{\log_{10} 2}$   $\ans{\displaystyle{0.301}}$

    2. $\displaystyle{\log_{10} 64}$   $\ans{\displaystyle{1.806}}$

    3. $\displaystyle{\log_{10} \sqrt{40}}$   $\ans{\displaystyle{0.801}}$

    4. $\displaystyle{\log_{10} \sqrt[3]{5}}$   $\ans{\displaystyle{0.233}}$

    5. $\displaystyle{\log_{10} 0.8}$   $\ans{\displaystyle{-0.0969}}$

  12. Use the approximate values $\ln 2 \doteq 0.6931$ and $\ln 3 \doteq 1.0986$ to compute:

    1. $\displaystyle{\ln 6}$   $\ans{\displaystyle{1.792}}$

    2. $\displaystyle{\ln 8}$   $\ans{\displaystyle{2.079}}$

    3. $\displaystyle{\ln 2e^3}$   $\ans{\displaystyle{3.693}}$

    4. $\displaystyle{\ln \frac{4}{3}}$   $\ans{\displaystyle{0.2877}}$

    5. $\displaystyle{\ln \frac{e}{9}}$   $\ans{\displaystyle{-1.197}}$

    6. $\displaystyle{\ln \frac{1}{27\sqrt{e}}}$   $\ans{\displaystyle{-3.796}}$

  13. Simplify and write as one logarithm:

    1. $\displaystyle{\log_{10} 2 + \log_{10} 5}$   $\ans{\displaystyle{\log_{10} 10 \quad \ldots\textrm{which equals } 1}}$

    2. $\displaystyle{\log_{10} (x^4 - 4) - \log_{10} (x^2 + 2)}$   $\ans{\displaystyle{\log_{10}(x^2-2)}}$

    3. $\displaystyle{3\,\ln 5 - \frac{1}{2}\,\ln 4 + \ln 8}$   $\ans{\displaystyle{\ln 500}}$

    4. $\displaystyle{\log_2 5 + \log_2 5^2 + \log_2 5^3 - \log_2 5^6}$   $\ans{\displaystyle{\log_2 1 \quad \ldots\textrm{which equals } 0}}$

    5. $\displaystyle{\ln (x^3 + 8) - \ln (x^2-2x+4) - 2\,\ln (x+2)}$   $\ans{\displaystyle{-\ln (x+2)}}$