Properties of Logarithms: Practice Exercises

Here are various Exercises to accompany the section Properties of Logarithms.  

Two Special Logarithms

1. Use your calculator to compute the following values.  Check your answer by plugging into the corresponding exponential function.
   1. log(8213)
   2. log(0.05217)
   3. ln(4.148)
   4. ln(1.321)
   5. log(-275)

   Solution.

Laws of Exponents and Logarithms

1. Use your calculator, and the change-of-base formula, with either the natural or common logarithm, to compute the following values.  
   1. log_0.75(4)
   2. log_0.5(0.35)
   3. log_0.49(0.343)
   4. log₂(5)
   5. Fill-in-the-blanks to create your own Exercise: log_{_}(__).

      Repeat this Exercise as often as necessary until you are confident in your ability to use the change-of-base formula.

   Solution.

Solving Exponential Equations

1. For each of the following pairs of equations, either:   
   • Determine which rule of logarithms was used to go from the first equation to the second, and how it was used (i.e., match up the variables in the formula for the rule and the corresponding expressions in the equation), or
   • Explain how the step misused a rule of logarithms.
   1. -7 = 4 - 5 log₃(3^x)    Þ    -7 = 4 - 5x
   2. -7 = 4 - 5 log₃(2^x)    Þ    -7 = 4 - 5x
   3. -7 = 4 - 5 log₃(2^x)    Þ    -7 = 4 - 5xlog₃(2)
   4. 3 = log₂(5·2^x)    Þ    3 = x log₂(5·2)
   5. 3 = log₂(5·2^x)    Þ    3 = log₂(5) + log₂(2^x)
   6. 2 = log₇(5/x)    Þ    2 = log₇(5)/ log₇(x)
   7. 2 = 4log₇(5/x)    Þ    2 = 4log₇(5) - log₇(x)
   8. Have your partner create a similar Exercise, by either correctly or incorrectly applying a rule of logarithms to some expression.

      Repeat this Exercise as often as necessary until you are confident in your ability to correctly apply the rules of logarithms.

   Solution.

2. Use algebra and rules of logarithms to solve the following exponential equations.  Give your answer to at least 6 correct digits.  Check your answer by plugging back into the equation.

   1. 191 = 3^x/9.72 - 416
   2. 3^{(x + 1)} = 26·2^-x
   3. 3·2^{(x - 2)/7} - 35 = 1501
   4. 21·2^{(x + 1)}3^{(1 - x)} = 56
   5. Have your partner create a similar Exercise.

      Repeat this Exercise as often as necessary until you are confident in your ability to correctly apply the rules of logarithms to solve exponential equations.

   Solution.

Go to Exponential Models and Logarithms .

Table of Contents

Send questions or comments to jwicks@northpark.edu

Glossary