Laws of Logarithms (OCR AS Maths: Pure)

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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

Laws of Logarithms Example fig1, A Level & AS Level Pure Maths Revision Notes

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"ln"

What is ln? 

  • ln is a function that stands for natural logarithm
  • It is a logarithm where the base is the constant "e"
    • ln space x identical to log subscript straight e x
    • It is important to remember that ln is a function and not a number

What are the properties of ln? 

  • Using the definition of a logarithm you can see
    • ln space 1 equals 0
    • ln space straight e equals 1
    • ln space straight e to the power of x equals x 
    • ln space x is only defined for positive x
  • As ln is a logarithm you can use the laws of logarithms
    • ln space a plus ln space b equals ln left parenthesis a b right parenthesis
    • ln space a minus ln space b equals ln stretchy left parenthesis a over b stretchy right parenthesis
    • n space ln space a equals ln left parenthesis a to the power of n right parenthesis

How can I solve equations involving e & ln? 

  • The functions straight e to the power of x and ln space x are inverses of each other
    • If straight e to the power of straight f left parenthesis x right parenthesis end exponent equals straight g left parenthesis x right parenthesis then straight f stretchy left parenthesis x stretchy right parenthesis equals ln invisible function application space straight g stretchy left parenthesis x stretchy right parenthesis
    • If ln invisible function application space straight f left parenthesis x right parenthesis equals straight g open parentheses x close parentheses then straight f open parentheses x close parentheses equals straight e to the power of straight g open parentheses x close parentheses end exponent
  • If your equation involves "e" then try to get all the "e" terms on one side
    • If "e" terms are multiplied, you can add the powers
      • straight e to the power of x cross times straight e to the power of y equals straight e to the power of x plus y end exponent 
      • You can then apply ln to both sides of the equation
    • If "e" terms are added, try transforming the equation with a substitution
      • For example: If y equals straight e to the power of x then straight e to the power of 4 x end exponent equals y to the power of 4
      • You can then solve the resulting equation (usually a quadratic)
      • Once you solve for y then solve for x using the substitution formula
  • If your equation involves "ln", try to combine all "ln" terms together
    • Use the laws of logarithms to combine terms into a single term
    • If you have ln invisible function application space straight f open parentheses x close parentheses equals ln invisible function application space straight g left parenthesis x right parenthesis then solve straight f open parentheses x close parentheses equals straight g left parenthesis x right parenthesis
    • If you have ln invisible function application space straight f open parentheses x close parentheses equals k then solve straight f open parentheses x close parentheses equals straight e to the power of k

Worked example

3-1-1-ln-we-solution

Examiner Tip

  • Always simplify your answer if you can
    • for example, 1 half ln space 25 space equals space ln space square root of 25 equals ln space 5
    • you wouldn't leave your final answer as square root of 25 so don't leave your final answer as 1 half ln space 25

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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.