Logarithmic Functions (Cambridge (CIE) AS Maths): Revision Note

Logarithmic functions

What are logarithmic functions?

• a = b^x and log _b a = x are equivalent statements

• a > 0

• b is called the base

• Every time you write a logarithm statement say to yourself what it means

  • log₃ 81 = 4

    “the power you raise 3 to, to get 81, is 4”

  • log_p q = r

    “the power you raise p to, to get q, is r”

How are logarithms and powers inverses of each other?

• A logarithm is the inverse of raising to a power so we can use rules to simplify logarithmic functions

  

How do I simplify expressions using the inverse property of logs and powers?

•  Recognising the rules of logarithms allows expressions to be simplified

• Recognition of common powers helps in simple cases

  • Powers of 2: 2⁰ = 1, 2¹ = 2, 2² = 4, 2³ = 8, 2⁴ =16, …

  • Powers of 3: 3⁰ = 1, 3¹ = 3, 3² = 9, 3³ = 27, 3⁴ = 81, …

  • The first few powers of 4, 5 and 10 should also be familiar

  For more awkward cases a calculator is needed 

  

• Calculators can have, possibly, three different logarithm buttons

• This button allows you to type in any number for the base

• Natural logarithms (see “e”)

• Shortcut for base 10 although SHIFT button needed 

• Before calculators, logarithmic values had to be looked up in printed tables

What notation is used with logarithms?

• 10 is a common base

  • log₁₀ x is abbreviated to log x or lg x

• The value e is another common base

  • log_e x is abbreviated to ln x

• (log x)² ≠ log x²