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ASMaths: Pure 2CIERevision Notes2. Logs & Exponentials2.2 Laws of Logarithms2.2.1 Laws of Logarithms

Laws of Logarithms (CIE AS Maths: Pure 2)

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Laws of Logarithms

What are the laws of logarithms?

Laws of Logarithms Notes fig1, A Level & AS Maths: Pure revision notes

 

  • There are many laws or rules of indices, for example
    • am x an = am+n
    • (am)n = amn

  • There are equivalent laws of logarithms (for a > 0)
    • log subscript a x y equals log subscript a x plus log subscript a y
    • log subscript a stretchy left parenthesis x over y stretchy right parenthesis equals log subscript a x minus log subscript a y
    • log subscript a x to the power of k equals k space log subscript a x 

Laws of Logarithms Notes fig2, A Level & AS Level Pure Maths Revision Notes

 

  • There are also some particular results these lead to
    • log subscript a a equals 1
    • log subscript a a to the power of x equals x
    • a to the power of log subscript a x end exponent equals x
    • log subscript a 1 equals 0
    • log subscript a stretchy left parenthesis 1 over x stretchy right parenthesis equals negative log subscript a x

Laws of Logarithms Notes fig3, A Level & AS Level Pure Maths Revision Notes 

  • Two of these were seen in the notes Logarithmic Functions
  • Beware …
    • … log (x + y) ≠ log x + log y
  • Results apply to ln too
    • ln space x space identical to log subscript straight e x
    • In particular straight e to the power of ln space x end exponent equals x and ln left parenthesis straight e to the power of x right parenthesis equals x

How do I use the laws of logarithms?

  • Laws of logarithms can be used to …
    • … simplify expressions
    • … solve logarithmic equations
    • … solve exponential equations

    Laws of Logarithms Notes fig4, A Level & AS Level Pure Maths Revision Notes

Examiner Tip

  • Remember to check whether your solutions are valid. log (x+k) is only defined if x > -k.
  • You will lose marks if you forget to reject invalid solutions.

Worked example

2-2-1-laws-of-logs-we-solution-1

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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.