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

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
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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
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:
- ASUFirst Year Math "Laws" of Logarithmic Functions lecture notes
- Oak RoadsSystems: It's the Law Too -- the Laws of Logarithms
- The MathPage: Skill in Algebra - Logarithms
- BBCEducation - AS Guru - Maths - Pure - Exponentials and Logarithms
- Lawsof Logarithms (this page contains several examples with answersavailable by clicking on the number)
- Mathcentre- The Laws of Logarithms
- Laws ofLogarithms
- Purplemath-Your Algebra Resource: Logarithm Rules I
- Purplemath-Your Algebra Resource: Logarithm Rules II
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".
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.