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Laws of Logarithmic Functions

Lawsof LogarithmicFunctions


When you are done with thissection, you will be able to do the following

  • Usethe change of base formula to evaluate a logarithm.
  • Applyproperties of logarithms.

There are several "Laws" of LogarithmicFunctions that will help us in solving logarithmic equations.

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


2.  Division becomes Subtraction

3.  Exponent becomes Multiplier

When you use the "Laws" to make onelogarithm into multiple logarithms, it is called expanding thelogarithm.  One common set of instructions would say to "rewritethe 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 singlelogarithm".  We can refer to the "opposite" direction with thefollowing names

1.  Addition becomes Multiplication

2.  Subtraction becomes Division

3.  Multiplier becomes Exponent

 

Another formula related tologarithms will allo w us to use our calculators to calculate logarithmsof any base.  If you look at your calculator, you can onlycalculate log (log base 10) and ln (natural log or log base e). In the last section, we were able to calculate logarithms with otherbases such as using thedefinition of a logarithm (the answer is 3).  What we don't knowhow to deal with is something like .  Theformula called the "change of base" formula will allow us to find adecimal value for .

Change of Base Formula


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

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

We could also choose to change to base e.  This would giveus c = 2 , a = 9, andb = e.  We would get .  Youwill see that when you plug this into your calculator, you will get thesame decimal answers as with the change to base 10.


A good place to find information about "Laws" oflogarithmic functionsisyour text book (pages 407-411) in CollegeAlgebra 3rd edition by Stewart, Redlin,and Watson).


Read carefully:
    Be aware of somecommon mistakes in applying the "Laws" of logarithms.


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


Laws of LogarithmicFunctions Homework:

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

  • Alwaysbe aware of which property you areusing.
    • Whenwriting the answers, the logarithm is usually written and you only needto fill in numbers and variables.  You do not need to type thework "log".

    Homeworkexamples:
    also-recommendedHomeworkexamples:
      • E xample6:  similar to problem 1 also-recommended
      • Example7:  may be helpful for problem 2 also-recommended
      • Example8:  similar to problem 3 also-recommended
      • Example9:  similar to problem 4 also-recommended

    Keep in mind thatitis recommendedthat youalso complete all theproblems in the set called "alsorecommended" with the same number as the Laws of Logarit hmic FunctionsHomework. Therearemany different ways to ask the same questions.  This will allowyou to see additional problems that are related to this topic.

    © 2005 Elizabeth E. K. Jones and the ASUDepartment ofMathematics and Statistics - All rights reserved.