Logarithmic Functions (Cambridge (CIE) A Level Maths) : Revision Note

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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 functions

    Logarithmic 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 Content Creator (Previous)

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.

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