Laws & Change of Base (Edexcel International A Level Maths): Revision Note

Laws of Logarithms

What are the laws of logarithms?

• There are many laws or rules of indices, for example

  • a^m x aⁿ = a^m+n

  • (a^m)ⁿ = a^mn

• There are equivalent laws of logarithms (for a > 0)

  • 

  • 

  •  

• There are also some particular results these lead to

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  • 

  • 

  • 

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• Two of these were seen in the notes Logarithmic Functions

• Beware …

  • … log (x + y) ≠ log x + log y

How do I use the laws of logarithms?

• Laws of logarithms can be used to …

  • … simplify expressions

  • … solve logarithmic equations

  • … solve exponential equations

Change of Base

What is the change of base formula? 

• We can rewrite a logarithm as a multiple of a logarithm of any different positive base using the formula

What is the use of the change of base formula? 

• This formula had more use when calculators were less advanced

  • Some old calculators only had a button for logarithm of base 10

  • To calculate on these calculators you would have to enter

    • 

• The formula is only needed in a small number of cases

  • This is given in the formulae booklet in case it is needed

• The formula can be useful when evaluation a logarithm where the two numbers are powers of a common number

  • 

• The formula can be useful when you are solving equations and two logarithms have different bases

  • For example, if you have and  within the same equation

    • You can rewrite  as which simplifies to 

    • Or you can rewrite  as which simplifies to 

• The formula also allows you to derive and use a formula for switching the numbers:

• Using the fact that