College Algebra Tutorial 46


College Algebra
Tutorial 46: Logarithmic Equations



WTAMU > Virtual Math Lab > College Algebra

 

deskLearning Objectives



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




deskIntroduction



In this tutorial I will step you through how to solve equations that have logarithmic expressions.  In these equations, you will notice that the variable that we are solving for is inside the log expressions.  We will rewrite the log equation exponentially using the definition of logs to help us get the x outside of the log.  If you need a review on the definition of log functions, feel free to go to Tutorial 43: Logarithmic Functions.  Ready, set, GO!!!!!

 

 

desk Tutorial


 
 
  Solving a Logarithmic Equation of the Form 

log
 

Step 1:  Write as one log isolated on one side.

 
Get your log on one side everything outside of the log on the other side of your equation using inverse operations. 

Also use properties of logs to write it so that there is only one log.  If you need a review on properties of logs feel free to go to Tutorial 44: Logarithmic Properties.
 

Step 2: Use the definition of logarithms to write in exponential form.

 
A reminder that the definition of logarithms is the logarithmic function with base b, where b > 0 and bnot equal0, and is defined as log if and only if log.

If you need a review on the definition of log functions, feel free to go to Tutorial 43: Logarithmic Functions.
 

Step 3: Solve for x.

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

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

 
Step 1:  Write as one log isolated on one side.

 
This is already done for us in this problem.

 
Step 2: Use the definition of logarithms to write in exponential form.

  example 1b
*Rewrite in exponential form
*Base = 5 and exponent = 3

 
Step 3: Solve for x.

 
example 1c

*5 cubed is 125
*Solve for x

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

 
Step 1:  Write as one log isolated on one side.

 
example 2b

*Use the product rule to write as one log
*one log is isolated

 
Step 2: Use the definition of logarithms to write in exponential form.

 
example 2c
*Rewrite in exponential form
*Base = 10 and exponent = 2

 
Remember that when there is no base written on a log, that means it is the common log, or log base 10.  If you need a review on common logs feel free to go to Tutorial 43: Logarithmic Functions.

 
Step 3: Solve for x.

 
example 2d

*10 squared is 100
*Solve for x

*Factor the trinomial

*Set 1st factor = 0 
 
 
 
 

*Set 2nd factor = 0 

 
 

Since -25 would create a negative number inside both logs in this problem and we CANNOT take the log of a negative number, we will have to throw out x = -25 as one of our solutions.
 

Final answer: x = 4.
 
 
 

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

 
Step 1:  Write as one log isolated on one side.

 
example 3b

*Use the quotient rule to write as one log
*one log is isolated

 
Step 2: Use the definition of logarithms to write in exponential form.

 
example 3c
*Rewrite in exponential form
*Base = 3 and exponent = 2

 
Step 3: Solve for x.

 
example 3d

*3 squared is 9
*Solve for x
 

*Multiply both sides by LCD of x + 2
 
 
 
 
 
 
 
 
 

 
 

The equation that we had to solve in step 3 had a rational expression in it.  If you need a review on solving equations with rational expressions feel free to go to Tutorial 15: Equations with Rational Expressions.

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

 
Step 1:  Write as one log isolated on one side.

 
example 4b

*Inverse of add 3 is sub. 3

*one log is isolated
 

Step 2: Use the definition of logarithms to write in exponential form.

 
example 4c
*Rewrite in exponential form
*Base = e and exponent = 1

 
Remember that when you have ln, that means it is the natural log, or log base e.  If you need a review on natural logs feel free to go to Tutorial 43: Logarithmic Functions.

 
 
Step 3: Solve for x.

 
example 4d

*Square both sides to get rid of the radical

*Solve for x
 

*Use the calculator to find e squared
 

 
 

The equation that we had to solve in step 3 had a square root in it.  If you need a review on solving equations with radicals feel free to go to Tutorial 19: Radical Equations and Equations Involving Rational Exponents.

 

 
 

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 - 1d:  Solve the logarithmic equation.  Round your answer to two decimal places.

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

 

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

 

 

 

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/solvelog.htm
This webpage will help you with solving logarithmic equations.

http://www.sosmath.com/algebra/solve/solve8/s81/s81.html
This webpage gives an example of solving a logarithmic 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.