Review - Exponential and Logarithmic Functions

  1. Find the value:

    1. $\displaystyle{\log_{32} \sqrt[4]{8}}$   $\ans{\displaystyle{\frac{3}{20}}}$

    2. $\displaystyle{\log_9 \sqrt[4]{\frac{1}{27}}}$   $\ans{\displaystyle{-\frac{3}{8}}}$

    3. $\displaystyle{\log_{1/5} \sqrt[7]{125^3}}$   $\ans{\displaystyle{-\frac{9}{7}}}$

    4. $\displaystyle{\log_{1/4} \sqrt[6]{64}}$   $\ans{\displaystyle{-\frac{1}{2}}}$

    5. $\displaystyle{\ln \left( \frac{\sqrt{e^3}}{e} \right)}$   $\ans{\displaystyle{\frac{1}{2}}}$

    6. $\displaystyle{e^{-\ln 3}}$   $\ans{\displaystyle{\frac{1}{3}}}$

  2. Simplify:

    1. $\displaystyle{\frac{2}{3} \log_{10} 125 - \frac{1}{2} \log_{10} 81 + \log_{10} \frac{18}{5}}$   $\ans{\displaystyle{1}}$

    2. $\displaystyle{\left( e^{-\ln 3} \right) \left( e^{4\ln \sqrt{3} - \frac{3}{2} \ln 16} \right) + \left( \ln e^{3/4} \right)^2}$   $\ans{\displaystyle{\frac{39}{64}}}$

    3. $\displaystyle{\log_7 48 + \log_7 21 - \log_7 9 - 2 \log_7 4}$   $\ans{\displaystyle{1}}$

    4. $\displaystyle{e^{\frac{1}{2} \ln \frac{1}{16} \, - \, \frac{2}{3} \ln 27} \, - \, \ln e^{5/4}}$   $\ans{\displaystyle{-\frac{11}{9}}}$

    5. $\displaystyle{\log_5 \sqrt[3]{\frac{1}{625}} + \log_3 27 - \log_2 8 + \log_{\pi} 1}$   $\ans{\displaystyle{-\frac{4}{3}}}$

    6. $\displaystyle{8^{\log_2 3}}$   $\ans{\displaystyle{27}}$

  3. Solve for $x$:

    1. $\displaystyle{1+2\log_3 x = \log_3 (1-x) + \log_3 (1+x)}$   $\ans{\displaystyle{\frac{1}{2} \textrm{ only}}}$

    2. $\displaystyle{(\ln x)^2 = \ln x^2}$   $\ans{\displaystyle{1,e^2}}$

    3. $\displaystyle{\log_x 36 = 2}$   $\ans{\displaystyle{6 \textrm{ only}}}$

    4. $\displaystyle{\log_{3/5} x = -3}$   $\ans{\displaystyle{\frac{125}{27}}}$

    5. $\displaystyle{4^x + 1 = 6(4^{-x})}$   $\ans{\displaystyle{\frac{1}{2}}}$

    6. $\displaystyle{\log_2 (x-2) + \log_2 (x-2) = 2}$   $\ans{\displaystyle{4}}$

    7. $\displaystyle{\log_3 (x^2-6x) = 3}$   $\ans{\displaystyle{-3,9}}$

    8. $\displaystyle{3^{x+1} = 27^{2x-3}}$   $\ans{\displaystyle{2}}$

    9. $\displaystyle{\log_x 9 = -\frac{2}{3}}$   $\ans{\displaystyle{\frac{1}{27}}}$

    10. $\displaystyle{e^{-\frac{1}{2} \ln x} = 4}$   $\ans{\displaystyle{\frac{1}{16}}}$

    11. $\displaystyle{\frac{1}{2} \log_4 (x+4) = 1}$   $\ans{\displaystyle{12}}$

    12. $\displaystyle{\ln x^3 = \frac{1}{e}}$   $\ans{\displaystyle{e^{\frac{1}{3e}}}}$

    13. $\displaystyle{\left( \frac{1}{2} \right)^{2x+1} = 8}$   $\ans{\displaystyle{-2}}$

    14. $\displaystyle{\log_3 (x-2) + \log_3 (x+3) = \log_3 6}$   $\ans{\displaystyle{3 \textrm{ only}}}$

    15. $\displaystyle{\left[ \log_3 (x-2) \right]^2 + 3\log_3 (x-2) = 0}$   $\ans{\displaystyle{3, \frac{55}{27}}}$

    16. $\displaystyle{\ln x = \ln 1 + \ln 2 + \ln 3 + \ln 4}$   $\ans{\displaystyle{24}}$

    17. $\displaystyle{\log_5 (3x+4)^{1/2} - \log_5 \sqrt{x} = 5^{\log_5 1}}$   $\ans{\displaystyle{\frac{2}{11}}}$

    18. $\displaystyle{\log_6 (x+3) + \log_6 (x+4) = \log_6 6}$   $\ans{\displaystyle{-1 \textrm{ only}}}$

    19. $\displaystyle{\frac{1}{6^x} = \sqrt[3]{36}}$   $\ans{\displaystyle{-\frac{2}{3}}}$

    20. $\displaystyle{4^{2x} - 5(4^x) + 4 = 0}$   $\ans{\displaystyle{0,1}}$

    21. $\displaystyle{2^{x+1} = 4^{\frac{3}{2}x - 1}}$   $\ans{\displaystyle{\frac{3}{2}}}$

    22. $\displaystyle{2^{x^2-5x+6} = 1}$   $\ans{\displaystyle{3,2}}$

    23. $\displaystyle{\log_2 (x-7) + \log_2 x = \ln e^3}$   $\ans{\displaystyle{8 \textrm{ only}}}$

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

  4. Sketch graphs of the following. Label intercepts and asymptotes. Give domain and range

    1. $\displaystyle{y=2^{x+1} - 4}$   $\ians{}$

    2. $\displaystyle{y = \log_3 (x-2) - 1}$   $\ians{}$

    3. $\displaystyle{y = 2^{-x} - 2}$   $\ians{}$

    4. $\displaystyle{y = \log_4 (x+4)}$   $\ians{}$

    5. $\displaystyle{y = 3^{x-1}}$   $\ians{}$

    6. $\displaystyle{y = \log_2 (x+1)}$   $\ians{}$

    7. $\displaystyle{y = \log_3 (x-3)}$   $\ians{}$