Tuesday, February 3, 2015

Properties of Logarithms

Properties of Logarithms 

Change-of-Base 

Since most calculators only have two types of log keys, one for common logarithm (base 10) and one for natural logarithms (base e), you may use the Change-of-Base Formula.

If a, b, and x are positive real numbers and  and  .
Then  can be converted to a different base using any of the following formulas:

  • Base b
  • Base 10
  • Base e







Changing Bases Using Common Logarithms





















Changing Bases Using Natural Logarithms


















The Properties of Logarithms 

Let a be a positive number such that  and let n be a real number.  If u and v are positive real numbers, the following properties are true:

  •                     
         Example:

         
         


  •                          

         Example:

         
         


  •                                    

         Example:

         
         

Rewriting Logarithmic Expressions 

To expand a logarithmic expression you must use the logarithmic properties to make the expression as "stretched out" as possible.

Expanding the Logarithm of a Product










Expanding the Logarithm of a Quotient 
















 To condense a logarithmic expression you must make the expression as simplified as possible by using the properties.

Condensing a Logarithmic Expression

 
     








Works Cited

Larson, Ron, Robert P. Hostetler, Bruce H. Edwards, and David E. Heyd.Precalculus with Limits: A Graphing Approach. Boston: Houghton Mifflin, 1997. Print.

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