• Use the change of base formula to evaluate a logarithm.
• Apply properties of logarithms.

The "Laws" of Logarithms will allow us to change a single logarithm into the sum, difference, and/or production of logarithms.  It is often easier to refer to the "Laws" if we give them names.  The are not standard names used in textbooks.
1.  Multiplication becomes Addition

2.  Division becomes Subtraction

3.  Exponent becomes Multiplier

When you use the "Laws" to make one logarithm into multiple logarithms, it is called expanding the logarithm.  One common set of instructions would say to "rewrite the logarithmic expression in a form with no logarithm of a product, quotient, or power".


The "Laws" of logarithms can also be used in the "opposite" direction.  This might be called condensing the logarithm.  The standard instructions would say "rewrite the expression as a single logarithm".  We can refer to the "opposite" direction with the following names

1.  Addition becomes Multiplication

2.  Subtraction becomes Division

3.  Multiplier becomes Exponent

Another formula related to logarithms will allow us to use our calculators to calculate logarithms of any base.  If you look at your calculator, you can only calculate log (log base 10) and ln (natural log or log base e).  In the last section, we were able to calculate logarithms with other bases such as using the definition of a logarithm (the answer is 3).  What we don't know how to deal with is something like .  The formula called the "change of base" formula will allow us to find a decimal value for .

Change of Base Formula

b is the new base that we are changing it to.  For our purposes (determining a decimal value), this new base can be either 10 or e. 

For our particular example , c = 2 and a = 9.  We will first change to base 10.  So b will be 10.  This will give us .  Plug this into your calculator to find the decimal value.

We could also choose to change to base e.  This would give us c = 2 , a = 9, and b = e.  We would get .  You will see that when you plug this into your calculator, you will get the same decimal answers as with the change to base 10.



A good place to find information about "Laws" of logarithmic functions is your text book (pages 407-411) in College Algebra 3rd edition by Stewart, Redlin, and Watson).

Read carefully:

Be aware of some common mistakes in applying the "Laws" of logarithms.

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Additional "Laws" of Logarithmic Functions On-line Resources:

• ASU First Year Math "Laws" of Logarithmic Functions lecture notes
• Oak Roads Systems:  It's the Law Too -- the Laws of Logarithms
• The Math Page:  Skill in Algebra - Logarithms
• BBC Education - AS Guru - Maths - Pure - Exponentials and Logarithms
• Laws of Logarithms (this page contains several examples with answers available by clicking on the number)
• Mathcentre - The Laws of Logarithms
• Laws of Logarithms
• Purplemath - Your Algebra Resource:  Logarithm Rules I
• Purplemath - Your Algebra Resource:  Logarithm Rules II
• Purplemath - Your Algebra Resource:  Logarithm Rules III

Laws of Logarithmic Functions Homework:

The Laws of Logarithmic Functions Homework contains 5 problems.
Things to remember:

• Always be aware of which property you are using.
• When writing the answers, the logarithm is usually written and you only need to fill in numbers and variables.  You do not need to type the work "log".
  

Homework examples:

• Example 1:  similar to problem 1
• Example 2:  may be helpful for problem 2
• Example 3:  similar to problem 3
  
• Example 4:  similar to problem 4
  
• Example 5:  may be generally helpful
  

also-recommended Homework examples:

• Example 6:  similar to problem 1 also-recommended
• Example 7:  may be helpful for problem 2 also-recommended
• Example 8:  similar to problem 3 also-recommended
• Example 9:  similar to problem 4 also-recommended

Keep in mind that it is recommended that you also complete all the problems in the set called "also recommended" with the same number as the Laws of Logarithmic Functions Homework.  There are many different ways to ask the same questions.  This will allow you to see additional problems that are related to this topic.

© 2005 Elizabeth E. K. Jones and the ASU Department of Mathematics and Statistics - All rights reserved.