Revision NotesExam QuestionsPast Papers
A LevelMathsOCRMaths APureRevision NotesExponentials & LogarithmsLaws of LogarithmsLaws of Logarithms

Laws of Logarithms (OCR A Level Maths A) : Revision Note

Paul

Author

Paul

Last updated

OCR Maths A: PureAQA Maths: PureEdexcel Maths: Pure

Did this video help you?

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 Tips and Tricks

  • 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

Did this video help you?

"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 Tips and Tricks

  • 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

You've read 0 of your 5 free revision notes this week

Sign up now. It’s free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

    Test yourself

    Did this page help you?

    Previous:Derivatives of Exponential FunctionsNext:Exponential Equations
    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.