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Laws of Logarithms (DP IB Analysis & Approaches (AA)): Revision Note

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Laws of Logarithms

What are the laws of logarithms?

  • Laws of logarithms allow you to simplify and manipulate expressions involving logarithms

    • The laws of logarithms are equivalent to the laws of indices

  • The laws you need to know are, given a comma space x comma space y space greater than space 0:

    • log subscript a x y equals blank log subscript a x plus blank log subscript a y

      • This relates to a to the power of x cross times blank a to the power of y equals a to the power of x plus y end exponent

    • log subscript a x over y equals blank log subscript a x blank negative space log subscript a y 

      • This relates to a to the power of x divided by blank a to the power of y equals a to the power of x minus y end exponent

    • log subscript a x to the power of m equals blank m log subscript a x 

      • This relates to left parenthesis a to the power of x right parenthesis to the power of y equals a to the power of x y end exponent

  • These laws are in the formula booklet so you do not need to remember them

    • You must make sure you know how to use them

Laws of Logarithms Notes fig2

Useful results from the laws of logarithms

  • Given a space greater than space 0 space comma space a space not equal to space 1

    • log subscript a 1 equals blank 0

      • This is equivalent to a to the power of 0 equals 1

  • If we substitute b for a into the given identity in the formula booklet

    • a to the power of x equals b space left right double arrow space log subscript a b space equals space x where a space greater than space 0 comma space b space greater than space 0 comma space a space not equal to space 1

    • a to the power of x space equals space a space left right double arrow space log subscript a a space equals space x gives a to the power of 1 space equals space a space left right double arrow space log subscript a a space equals space 1 

      • This is an important and useful result

  • Substituting this into the third law gives the result

    • log subscript a a to the power of k equals blank k

  • Taking the inverse of its operation gives the result

    • a to the power of log subscript a x end exponent equals blank x

  • From the third law we can also conclude that

    • log subscript a 1 over x equals blank minus log subscript a x

Laws of Logarithms Notes fig3

  • These useful results are not in the formula booklet but can be deduced from the laws that are

  • Beware…

    • …log subscript a open parentheses x plus y close parentheses space not equal to space log subscript a x plus log subscript a y

  • These results apply to ln space x space left parenthesis log subscript e x right parenthesis too

    • Two particularly useful results are

      • ln space e to the power of x space equals space x

      • e to the power of ln x end exponent space equals space x

  • Laws of logarithms can be used to …

    • simplify expressions

    • solve logarithmic equations

    • solve exponential equations

Examiner Tips and Tricks

  • Remember to check whether your solutions are valid

    • log (x+k) is only defined if x > -k

    • You will lose marks if you forget to reject invalid solutions

Worked Example

a) Write the expression 2 space log space 4 space minus space log space 2 in the form log space k, where k space element of space straight integer numbers.

 

aa-sl-1-2-2-laws-of-logs-we-solution-part-a

b)   Hence, or otherwise, solve 2 space log space 4 minus log space 2 equals negative log blank 1 over x.

aa-sl-1-2-2-laws-of-logs-we-solution-part-b

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Change of Base

Why change the base of a logarithm?

  • The laws of logarithms can only be used if the logs have the same base

    • If a problem involves logarithms with different bases, you can change the base of the logarithm and then apply the laws of logarithms

  • Changing the base of a logarithm can be particularly useful if you need to evaluate a log problem without a calculator

    • Choose the base such that you would know how to solve the problem from the equivalent exponent

How do I change the base of a logarithm?

  • The formula for changing the base of a logarithm is

begin mathsize 22px style log subscript a x equals blank fraction numerator log subscript b x over denominator log subscript b a end fraction end style

  • This is in the formula booklet

  • The value you choose for b does not matter, however if you do not have a calculator, you can choose b such that the problem will be possible to solve

Examiner Tips and Tricks

  • Changing the base is a key skill which can help you with many different types of questions, make sure you are confident with it

    • It is a particularly useful skill for examinations where a GDC is not permitted

Worked Example

By choosing a suitable value for b, use the change of base law to find the value of  log subscript 8 space end subscript 32 without using a calculator.

aa-sl-1-2-2-change-of-base-we-solution-1

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