Laws & Change of Base (Edexcel International A Level Maths: Pure 2)

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
  • 
  • 
  • 
  • 
  • 

  

• 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

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