Exercises


  1. Evaluate each logarithm without using a calculator.
    1. \( \log (100000) \)
    2. \(\log (0.001)\)
    3. \(\log (1)\)
    4. \(\log \left( \log_2 (1024) \right)\)
    5. \(\log_2 \left(\log \left(\sqrt{10} \right) \right)\)
    6. \(3^{\log_3(81)}\)
    7. \(10^{\log (0.0001)}\)
    8. \(6^{\log_6 (12)}\)
    9. \(4^{\log_{16} \left(\frac{1}{4}\right)}\)
  2. Identify the restrictions on \(x\). Convert each equation to the equivalent exponential form and solve for \(x\).
    1. \( \log_8 \left(\dfrac{1}{16\sqrt{2}}\right)= x\)
    2. \( \log_3 (4x+6)=2\)
    3. \( \log (3x-2)=1\)
    4. \( \log_x (16) = -2\)
    5. \( \log_x (64) = \frac{3}{2}\)
    6. \( \log_x (6-x)=2\)
  3. Using the law of logarithms, write each expression as a single logarithm in simplest form. State any restrictions on the variables.
    1. \( \log_2 (r) + \log_2 (s) - \log_2 (t) \)
    2. \( 3\log (a)-2\log (3b) \)
    3. \( \dfrac{\log(x)}{\log(y)} \)
    4. \( \dfrac{1}{2}\log_5(4x)-\log_5(x-3) \)
    5. \( \dfrac{1}{4}\Big(\log (y-1) + \log(x+1)\Big) \)
    6. \( 3\log_3 a -\log_3 b-\dfrac{1}{2}\log_3 c \)
  4. Simplify, then evaluate without using a calculator.
    1. \(\log_6 (18) + \log_6 (2)\)
    2. \(\log_2 (56) - \log_2 (7)\)
    3. \(\log (4) -4\log (2) - \log (25)\)
    4. \(5^{2\log_5 (6)}\)
    5. \(3\log_5 (10) - \log_5 (40)\)
    6. \(\log_4 (24)- 2\log_4 (3) + \dfrac{1}{2}\log_4 (144)\)
    7. \(\dfrac{\log_3 (216)}{\log_3 (6)}\)
    8. \(9^{\log_3(12)-\log_3(3)} \)
    9. \(3\log (200) - \dfrac{1}{2}\log (64) \)
  5. Express each of the following in terms of \( \log_2(a)\), \( \log_2(b)\), and/or \( \log_2(c)\), where \(a,b,c \gt 0\).
    1. \(\log_2(8a)\)
    2. \(\log_2(a^2b)\)
    3. \(\log_2 \left(\dfrac{\sqrt{a}}{bc}\right)\)
    4. \(\log_2(\frac{1}{2}ab)^5\)
    5. \(\log_2 \left(\dfrac{a^2\sqrt{b}}{c^3}\right)\)
    6. \(\log_2\left(\dfrac{2}{\sqrt[4]{8bc}}\right)\)
  6. Determine the value of each to three decimal places of accuracy.
    1. \(\log (185)\)
    2. \(\log_3 (65)\)
    3. \(\log_8 (5)\)
    4. \(2\log_{\frac{1}{4}} (24)\)
  7. Using logarithms, solve for \(x\) to three decimal places of accuracy.
    1. \(10^x=50\)
    2. \(3^x=35\)
    3. \((0.5)^{2x}=10\)
    4. \(2^{x-3}=18.5\)
    1. Show that \(\log_c (a) = \dfrac{1}{\log_a (c)}\), where \(a, c \gt 0\).
    2. Show that \(\log_a (b)\log_b(c)=\log_a (c)\), where \(a, b, c \gt 0\).
    1. The graph of \(f(x)=\log_2 \left(4x^3\right)\) can be obtained by transforming the graph of \(f(x)=\log_2 (x)\). Using the laws of logarithms, express \(f(x)=\log_2 \left(4x^3\right)\) in terms of \(\log_2 (x)\) and sketch its graph. Using graphing technology and the original version of the equation of the function, graph \(f(x)=\log_2 \left(4x^3\right)\). What do you notice?
    2. Using graphing technology, graph \(g(x)=\log_2 \left(x^2\right)\) and \(h(x)=2\log_2 (x)\). What do you notice?
    3. Can you explain why the laws of logarithms worked appropriately in part a) but not in part b)?
  10. Given \(f(x)=\dfrac{2^{\frac{1}{x}+1}}{1+2^{\frac{1}{x}+1}}\) and \(g(x)=\log_2\left(\dfrac{f(x)}{1-f(x)}\right)\), show that \(xg(x)-g\left(\dfrac{1}{x}\right)=0\) for all \(x\neq0, x \in\mathbb{R}\).