College Algebra Tutorial 45


College Algebra
Tutorial 45: Exponential Equations



WTAMU > Virtual Math Lab > College Algebra

 

deskLearning Objectives



After completing this tutorial, you should be able to:
  1. Solve exponential equations. 




deskIntroduction



In this tutorial I will walk you through how to solve equations that have exponential expressions.  In these equations, you will notice that the variable that we are solving for is in the exponent.  We are use to seeing the variable in the base.  We will using inverse operations like we do in linear equations, the inverse operation we will be using here is logarithms.  If you need a review on the definition of log functions, feel free to go to Tutorial 43: Logarithmic Functions.  If you need a review on log properties, feel free to go to Tutorial 44: Logarithmic Properties.  I think you are ready to get started.  

 

 

desk Tutorial


 
  Solving Exponential Equations, 
where x is in the exponent, BUT
the bases DO NOT MATCH.
 
 

Step 1: Isolate the exponential expression.

 
Get your exponential expression on one side everything outside of the exponential expression on the other side of your equation.

 
Step 2: Take the natural log of both sides.

 
The inverse operation of an exponential expression is a log.  Make sure that you do the same thing to both sides of your equation to keep them equal to each other.

 
Step 3: Use the properties of logs to pull the x out of the exponent.

 
If you need a review on log properties, feel free to go to Tutorial 44: Logarithmic Properties.  I think you are ready to get started.

 
Step 4: Solve for x.

 
Now that the variable is out of the exponent, solve for the variable using inverse operations to complete the problem.

 
 
 
Special Note:
The only way we can get that variable out of the exponent, when the bases don't match up, is to use logs.  The 3rd step allows us to do this.  When solving an equation, it doesn't matter what you do to the equation as long as you do the same thing to both sides - this keeps both sides equal.  Also, the reason we take the natural log of both sides is because we have the natural log key on the calculator - so we would be able to find a value of it in the end. 

 
 
 
notebook Example 1: Solve the exponential equation example 1a.  Round your answer to two decimal places.

 
 
Step 1: Isolate the exponential expression.

 
This is already done for us in this problem.

 
Step 2: Take the natural log of both sides.

 
example 1b
*Take the natural log of BOTH sides

 
Step 3: Use the properties of logs to pull the x out of the exponent.

 
example 1c
*Use the power rule

 
Step 4: Solve for x.

 
example 1d

*Inverse of mult. by 3lne is to divide by 3lne
 
 

*Use the calculator to find ln 50
*lne is 1
 

 
 
 
 

notebook Example 2: Solve the exponential equation example 2a.  Round your answer to two decimal places.

 
 
Step 1: Isolate the exponential expression.

 
example 2b

 

*Inverse of mult. by 5 is to divide by 5

*Exponential expression isolated
 
 

Step 2: Take the natural log of both sides.

 
example 2c
*Take the natural log of BOTH sides

 
Step 3: Use the properties of logs to pull the x out of the exponent.

 
example 2d
*Use the power rule

 
Step 4: Solve for x.

 
example 2e

*Inverse of mult. by ln 10 is to divide by ln 10
 
 
 

*Inverse of add 1 is sub. 1
 
 

*Use the calculator to find ln 2.4 and ln 10
 

 
 
 
 

notebook Example 3: Solve the exponential equation example 3a.  Round your answer to two decimal places.

 
 
Step 1: Isolate the exponential expression.

 
example 3b

*Inverse of add 4 is sub. 4

*Exponential expression isolated
 

Step 2: Take the natural log of both sides.

 
example 3c
*Take the natural log of BOTH sides

 
Step 3: Use the properties of logs to pull the x out of the exponent.

 
example 3d
*Use the power rule

 
Step 4: Solve for x.

 
example 3e

*Inverse of mult. by .2ln 2 is to divide by .2ln 2
 
 

*Use the calculator to find ln 21 and ln 2
 

 
 
 
 

notebook Example 4: Solve the exponential equation example 4h.  Round your answer to two decimal places.

 
 
Step 1: Isolate the exponential expression.

 
Notice how we have two exponential terms that have different exponents.  We wouldn't be able to isolate both.  We will have to figure out another way to rewrite it so we can continue with the steps. 

Note how we have a trinomial and that e to the 2x is e to the x squared.  This means it is quadratic in from.  So we can factor it just like a trinomial of the form example 4b.
 

example 4i

*Factor the trinomial of the form example 4b.
 

*Set the 1st factor = 0
*Isolate the exponential expression
 
 
 
 

*Set the 2nd factor = 0
*Isolate the exponential expression
 
 

Note that since e is a positive base, no matter what the exponent is on x, this exponential expression CANNOT equal -2.

So there is only one equation that we can solve example 4d.
 
 

Step 2: Take the natural log of both sides.

 
example 4e
*Take the natural log of BOTH sides

 
Step 3: Use the properties of logs to pull the x out of the exponent.

 
example 4f
*Use the power rule

 
Step 4: Solve for x.

 
example 4g
*Inverse of mult. by lne is to divide by lne
 
 
 

*Use the calculator to find ln 4
*lne = 1

 
 

 
 

desk Practice Problems



These are practice problems to help bring you to the next level.  It will allow you to check and see if you have an understanding of these types of problems. Math works just like anything else, if you want to get good at it, then you need to practice it.  Even the best athletes and musicians had help along the way and lots of practice, practice, practice, to get good at their sport or instrument.  In fact there is no such thing as too much practice.

To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that  problem.  At the link you will find the answer as well as any steps that went into finding that answer.

pencil Practice Problems 1a - 1c:  Solve the given exponential equation.  Round your answer to two decimal places.

 
1a. problem 1a
(answer/discussion to 1a)
1b. problem 1b
(answer/discussion to 1b)

 

1c. problem 1c
(answer/discussion to 1c)

 

 

 

desk Need Extra Help on these Topics?



The following are webpages that can assist you in the topics that were covered on this page.

http://www.purplemath.com/modules/solvexpo.htm
This webpage helps you with exponential equations.

http://www.sosmath.com/algebra/solve/solve7/s72/s72.html
This webpage gives an example of solving an exponential equation.

http://www.sosmath.com/algebra/solve/solve7/s73/s73.html
This webpage gives an example of solving an exponential equation.

 

Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.


 

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WTAMU > Virtual Math Lab > College Algebra


Last revised on March 24, 2011 by Kim Seward.
All contents copyright (C) 2002 - 2011, WTAMU and Kim Seward. All rights reserved.