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Section 3.2 – Logarithmic Functions and Their Graphs

Section Details.
  • Definition, properties, and terminology for logarithms
  • Evaluating logarithms
  • The natural log function
  • The relationship between logarithms and exponentials
  • Using the properties of logarithms to simplify a logarithmic expression
  • Graphing a logarithmic function and determining its properties


Practice Problems


Directions. The following are review problems for the section. We recommend you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it. 
 
  1. Simplify each of the following without a calculator:
    1. \(\log_{4} (64)\)
    2. \(7^{\log_{7} (4)}+2\)
    3. \(\displaystyle{\log (10^{-5})}\)
    4. \(\log_{11}(3x+5) =\log_{11} (9)\)

      a. \(\displaystyle{\log_{4} (64)}=3\)
      b. \(\displaystyle{7^{\log_{7} (4)}}+2=6\)
      c. \(\displaystyle{\log (10^{-5})}=-5\)
      d. \(\displaystyle{\log_{11}(3x+5) =\log_{11} (9)} \implies x=\dfrac{4}{3}\)


      To see the full video page and find related videos, click the following link.
      WIR4 20B M150 V6


  2. Consider the function \(g(x) = -\log_{3}(x+5)+2\).
    1. Describe the transformations of \(f(x) = \log_{3}(x)\) that yield \(g(x) = -\log_{3}(x+5)+2\). Then state the domain, \(x\)-intercept, \(y\)-intercept,  and vertical asymptote of the logarithmic function \(f(x)\).

      Transformations: Left 5, reflect over \(x\)-axis, up 2
      Domain: \((-5,\infty)\)
      \(x\)-intercept: \((4,0)\) 
      \(y\)-intercept: \((0,-\log_3(5)+2)\)
      Vertical asymptote: \(x=-5\)


      To see the full video page and find related videos, click the following link.
      WIR4 20B M150 V7


    2. Choose the graph that matches the function \(g(x) = -\log_{3}(x+5)+2\).

      The dark blue graph is the correct graph.


      To see the full video page and find related videos, click the following link.
      WIR4 20B M150 V8


  3. Consider the function \(g(x) =\ln (2x+3)-4\).
    1. Describe the transformations of \(f(x) = \ln (x)\) that yield \(g(x) =\ln (2x+3)-4\). Then state the domain, \(x\)-intercept, \(y\)-intercept, and vertical asymptote of the logarithmic function \(f(x)\).

      Transformations: Left 3/2, horizontal shrink by 1/2, down 4
      Domain: \(\left(-\dfrac{3}{2},\infty\right)\)
      \(x\)-intercept: \( \left(\dfrac{e^4-3}{2}, 0\right)\) 
      \(y\)-intercept: \((0,\ln(3)-4)\)
      Vertical asymptote: \(x=-\dfrac{3}{2}\)
       


      To see the full video page and find related videos, click the following link.
      WIR4 20B M150 V9


    2. Choose the graph that matches the function \(g(x) =\ln (2x+3)-4\).

      The dark blue graph is the correct graph.


      To see the full video page and find related videos, click the following link.
      WIR4 20B M150 V10