Introduction to Logarithms: Practice Exercises

Here are various Exercises to accompany the section Introduction to Logarithms.  

Introduction to Logarithmic Functions

1. Use the definition of logarithms to express the solutions to the following exponential equations:
   1. 20 = 5^x
   2. 4 = (0.75)^x
   3. 0.35 = (0.5)^x
   4. 0.343 = (0.49)^x
   5. -14 = -3·2^x + 1.  Hint: Solve the equation first to isolate the exponential, 2^x, then solve as before.

   Solution.

2. In the following applet, we have defined the functions f(x) = 5^x, s(x) = (0.75)^x, g(x) = (0.5)^x, h(x) = (0.49)^x, and k(x) = -3·2^x + 1:

   Use the "x = " box to estimate values, to three decimals, for the solutions (expressed in terms of logarithms) in the previous Exercise.

   Solution.

Graphing Logarithmic Functions

1. Create plots for each of the following pair of exponential and logarithmic functions:
   1. exp₃(x) = 3^x and log₃(x).
   2. exp_1/4(x) = (1/4)^x and log_1/4(x).
   3. Pick a base, b, and plot exp_b(x) = b^x and log_b(x) for your chosen base.  Repeat this Exercise as often as necessary until you are confident in your ability to plot logarithmic graphs.

   Solution.

2. Use our transformational graphing technique to graph the following functions with a logarithmic "core".  Hint: Start from the graphs from the previous Exercise:
   1. f(x) = 2log₃(-x + 1) - 4.
   2. g(x) = 3log_1/4(x - 2) + 1.
   3. Fill-in-the-blanks to create a formula for a function with a logarithmic "core", then use our transformational graphing technique to graph it.

      y = __·log_{_}(__x + __) + __

      Note: This includes choosing the base of the logarithm; you may want to use the same base that you chose in the previous Exercise.

      Repeat this Exercise as often as necessary until you are confident in your ability to plot functions with a logarithmic "core".

   Solution.

Go to Properties of Logarithms .

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