Logarithmic Functions (CIE A Level Maths: Pure 3)

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Logarithmic Functions

Logarithmic functions

Logarithmic Functions Notes fig1, A Level & AS Maths: Pure revision notes

  • a = bx 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”

    • logp q = r

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

Logarithm rules

  • A logarithm is the inverse of raising to a power so we can use rules to simplify logarithmic functionsLogarithmic Functions Notes fig2, A Level & AS Maths: Pure revision notes

How do I use logarithms?

Logarithmic Functions Notes fig3, A Level & AS Level Pure Maths Revision Notes

  •  Recognising the rules of logarithms allows expressions to be simplified

Logarithmic Functions Notes fig4, A Level & AS Maths: Pure revision notes

  • 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 Logarithmic Functions Notes fig5, A Level & AS Level Pure Maths Revision Notes
  • Calculators can have, possibly, three different logarithm buttons

Logarithmic Functions Notes fig6, A Level & AS Maths: Pure revision notes 

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

Logarithmic Functions Notes fig7, A Level & AS Maths: Pure revision notes

  • Natural logarithms (see “e”)

Logarithmic Functions Notes fig8, A Level & AS Maths: Pure revision notes

  • Shortcut for base 10 although SHIFT button needed 
  • Before calculators, logarithmic values had to be looked up in printed tables

Notation

Logarithmic Functions Notes fig9, A Level & AS Maths: Pure revision notes

  • 10 is a common base
    • log10 x is abbreviated to log x or lg x
  • The value e is another common base
    • loge x is abbreviated to ln x
  • (log x)2 ≠ log x2

Worked example

Logarithmic Functions Example fig1, A Level & AS Maths: Pure revision notes

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Paul

Author: Paul

Expertise: Maths

Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.