Lesson 6

Lesson 6: Solving Exponential Equations Using Logarithms

 

Focus

 

Now that you have learned the power law of logarithms, you can use the law to solve exponential equations. This means you can determine an exact value of t in equations like the E. coli growth equation you saw in Lesson 2. In this lesson you will learn how to solve exponential equations by applying the laws of logarithms.

 

Lesson Outcome

 

At the end of this lesson you will be able to solve exponential equations using logarithms.

 

Lesson Question

 

You will investigate the following question: How can logarithms be used to solve exponential equations?

 

Assessment

 

Your assessment may be based on a combination of the following tasks:

• completion of the Lesson 6 Assignment (Download the Lesson 6 Assignment and save it in your course folder now.)
• course folder submissions from Try This and Share activities
• work under Project Connection

Discover

 

In this activity you will determine the time when the bacteria reach a certain population using the equation of the laws of logarithms.

 

Try This 1

 

Suppose that you begin with a single E. coli bacterium at time 0, and the conditions are appropriate for the bacteria to double in population every 20 min. This growth can be modelled using the equation

1.  
   1. Create a table that shows the number of bacteria at 20-min intervals for 5 h. Your table might start out like this one.

       

      Time (in min)	Number of Bacteria
      0
      20
      40

   2. Use your table to estimate when there would be 10 000 bacteria.
2.  
   1. Follow the steps in the following table to algebraically determine an approximate time when there would be 10 000 bacteria. Make the assumption that the equation can be used to find an approximate time where there would be 10 000 bacteria.
      
      

      Write the equation.
      Substitute the known values for P and P0.
      Take the logarithm of both sides of the equation.
      Use the power law of logarithms, logb (Mn) = n logb M, to “bring down” the exponent
      Divide both sides of the equation by log 2.
      Multiply both sides of the equation by 20.
      Determine a decimal approximation of t.

   2. How does the time you determined in 2.a. compare to your estimate from 1.b.?

Save your responses in your course folder.

 

Share 1

 

With a partner or group, discuss the following based on your responses in Try This 1.

1. Explain why the power law of logarithms is necessary to solve many exponential equations.
2. How accurate do you expect the prediction you made in question 2.a. to be? Explain why.

If required, save a summary of your discussion in your course folder.

 

 

 

Explore

 

In Try This 1 you looked at a method for solving exponential equations using logarithms. Using the same conditions as in Try This 1, the time it takes for there to be 500 000 bacteria is determined by using the following equation. Notice when the power law of logarithms is used.

 

Just as with the other operations, you need to take the logarithm of an entire side of the equation when it is used. So, when given the equation 500 = 3(6)x, review the following table for correct use:

Incorrect Use of Logarithms	Correct Use of Logarithms

Also, when using the power law of logarithms, the logarithm must be of just the power. Continuing the example, consider the following table:

Incorrect Use of Power Law of Logarithms	Correct Use of Power Law of Logarithms

Sometimes an exponential equation will have more than one power in it. Solving these equations follows a process similar to the exponential equations you saw earlier. In Try This 2, you will explore how to solve this type of equation.

 

Try This 2

 

Open and complete Solving Exponential Equations 2.

 

Watch Solving Exponential Equations with Two Powers to see how an exponential equation with multiple powers, similar to the one you saw in Try This 2, can be solved.

 

 

When working with logarithms, it can be useful to change the base of the logarithm. Some calculators will not accept most bases, so converting to base 10 allows you to determine a decimal approximation. In Try This 3 you will explore a method that can be used to convert between bases.

 

Try This 3

1. Consider the equation y = log4 20. Complete the following steps to give an expression of y using base-10 logarithms.
   
   

   Write the original logarithm.	y = log4 20
   Rewrite the logarithm in exponential form.
   Take the base-10 logarithm of each side of the equation.
   Use the power law of logarithms, logb (Mn) = n logb M, to “bring down” the exponent y.
   Divide both sides by log 4 to isolate y.

2. Compare the resulting expression with the initial expression in question 1. Describe a rule showing how logb x can be rewritten using a different base.

Save your responses in your course folder.

In Try This 3, you may have found that the base of a logarithm can be changed using a formula equivalent to

 

Self-Check 3

1. Complete “Your Turn” at the bottom of page 453 of your textbook. Answer
2. Complete “Your Turn” on page 454 of your textbook. Answer
3. Complete questions 5.b. and 16.c. on pages 456 and 458 of your textbook. Answer

When a problem is modelled using an exponential equation, it is often useful to solve that equation using logarithms. In Try This 4, you will solve a problem involving compound interest.

 

Try This 4

 

When a bank sets a rate for an investment or a loan in Canada, the prime rate set by the Bank of Canada is often used as a guideline. In 1981, the prime rate reached 22.75%, and, in 2009, it dipped to 2.25%. Compare the time it takes an investment to increase using these two rates by completing the following questions.

1. Imagine depositing $1000 into a bank account that paid compound interest at 2.25% per year and $1000 into one that paid 22.75% per year. Write an exponential equation to represent the amount of money in each account after t years.
2. How much money would each account have after 5 years?
3. Determine how long it would take for each account to accumulate $5000 by solving the equations using logarithms.
4. Typically, extreme interest rates like these don’t last for very long. How would this change your expectation for the length of time taken for $1000 to grow into $5000?

Save your responses in your course folder.

In Try This 4 you may have found that it takes nearly 10 times as long for an investment at 2.25% to grow to 5 times its original amount than an investment at 22.75%. These rates are extreme, and most mortgage rates and savings-account rates would fall somewhere between them.

Connect

 

Lesson 6 Assignment

Lesson 6 Summary

 

In this lesson you explored a new way of solving an exponential equation that used logarithms. Taking the logarithm of each side of the equation and then using the power law for logarithms is a good way to solve exponential equations that cannot be solved easily using common bases.

 

Sometimes it is useful to change the base of a logarithm. The equation can be used to accomplish this.