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Exponential Equations

Approximating Solutions for Exponential Equations

Using a graphing utility

There are at least two different methods that can be used to approximate the solutions of an exponential equation using a graphing utility.

 
  1. Graph the left-hand side and the right-hand side of the equation in the same viewing window; then use the intersect feature or the zoom and trace features of the graphing utility to find the points of intersection.

  2. Rewrite the equation so that all terms on the left side are equal to 0. Then use a graphing utility to graph the left side of the equation. Use the zero or root feature or the zoom and trace features to approximate the solutions of the equation.

Example:  ex = 72

 

I. Using the first method, we will use a graphing utility to graph the left- and right-hand sides of the equation as y1 = ex and y2 = 72.

 
 

Using the intersect feature, the point of intersection of the two equations is (x, y) = (4.27666611901325, 71.99999999979796) is. Hence, an approximate solution is x≈ 4.28.

 
 

II. Using the second method, we will rewrite the equation such that all the terms on the left side are equal to 0. That is, ex  - 72 = 0. We now plot the equation y = ex  - 72 as shown below

 
 
 
 
 

Zooming in and using the zero or root feature of the graphing utility, we find x = 4.27667. Therefore, an approximate solution is x≈ 4.28.

 

Using the Exponential Table

Below is an illustration of how an exponential function table looks like.

 

x

ex

e-x

0.00

1.0000

1.000000

0.01

1.0101

0.990050

0.02

1.0202

0.980199

0.03

1.0305

0.970446

0.04

1.0408

0.960789


The exponential function table provides the approximate values of an exponential function. Thus, in order to find an approximate solution all you need to do is search through the table.

To find the approximate solution for the previous example ex = 72, we will use the following table.

x

ex

e-x

4.25

70.105

0.014264

4.26

70.810

0.014122

4.27

71.522

0.013982

4.28

72.240

0.013843

4.29

72.966

0.013705



Since ex = 72, we need to choose which value of ex from the table is nearest to our given equation. And so, we have three options, x = 4.27, 4.28, and 4.29.

Compare the values of ex .

If x = 4.27,       |72 – 71.522| = 0.4780

If x = 4.28,       |72 – 72.240| = 0.2400

If x = 4.29,       |72 – 72.966| = 0.9660

The value of ex that is closest to the given equation is 72.240. Thus, an approximate solution is x≈ 4.28.

 

Try these questions

1.
    Find an approximate solution:  5x = 26
     
  a. -2.02
  b. -2.03
  c. 2.02
  d. 2.03
     
2.
    The graph of y = 5x – 26 is
   
  a.
     
  b.
     
  c.
     
  d.
     
3.
    Find an approximate solution: e-4x = 0.231
     
  a.
      0.36
  b.
      0.37
  c.
      -0.36
  d.
      -0.37
     

ANSWERS TO PRACTICE TEST QUESTIONS

  1. C          2.02. Using a graphing utility it is easy to find the approximate solution for 5x = 26.


  2. A          Using a graphing utility it is easy to graph the function y = 5x – 26


  3. B          0.37. Use a graphing utility to find the approximate solution for e- 4x = 0.231 or use the exponential function for negative x table. First simplify: e- 4x = 0.231 ⇒ ( e-x )4= 0.231⇒( e-x ) = 0.69327. Using the table you will find that .690734 as the closest value and x = 0.37.
 
 

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