Revision NotesExam QuestionsFlashcardsPast Papers
IBMathsDPApplications & Interpretation (AI)HLRevision Notes1. Number & AlgebraExponentials & LogsLogarithms

Logarithms (DP IB Applications & Interpretation (AI)): Revision Note

Amber

Author

Amber

Last updated

DP Applications & Interpretation (AI): HLDP Analysis & Approaches (AA): HLDP Analysis & Approaches (AA): SL

Did this video help you?

Introduction to Logarithms

What are logarithms?

  • A logarithm is the inverse of an exponent

    • If a to the power of x equals b then log subscript a open parentheses b close parentheses equals x where a > 0, b > 0, a ≠ 1

      • This is in the formula booklet

      • The number a is called the base of the logarithm

      • Your GDC will be able to use this function to solve equations involving exponents

  • Try to get used to ‘reading’ logarithm statements to yourself

    • log subscript a left parenthesis b right parenthesis space equals space x would be read as “the power that you raise a to, to get b, is x”

    • So log subscript 5 125 space equals space 3 would be read as “the power that you raise 5 to, to get 125, is 3”

  • Two important cases are:

    • ln space x equals log subscript straight e open parentheses x close parentheses

      • Where e is the mathematical constant 2.718…

      • This is called the natural logarithm and will have its own button on your GDC

    • log space x equals log subscript 10 open parentheses x close parentheses

      • Logarithms of base 10 are used often and so abbreviated to log x

Why use logarithms?

  • Logarithms allow us to solve equations where the exponent is the unknown value

    • We can solve some of these by inspection

      • For example, for the equation 2x = 8 we know that x must be 3

    • Logarithms allow use to solve more complicated problems

      • For example, the equation 2x = 10 does not have a clear answer

      • Instead, we can use our GDCs to find the value of log subscript 2 10

Examiner Tips and Tricks

  • Before going into the exam, make sure you are completely familiar with your GDC and know how to use its logarithm functions

Worked Example

Solve the following equations:

i) x equals log subscript 3 27,

 

ai-sl-1-1-2intro-to-logs-we-i

ii) 2 to the power of x equals 21.4, giving your answer to 3 s.f.

 

ai-sl-1-1-2intro-to-logs-we-ii

Did this video help you?

Laws of Logarithms

What are the laws of logarithms?

  • Laws of logarithms allow you to simplify and manipulate expressions involving logarithms

    • The laws of logarithms are equivalent to the laws of indices

  • The laws you need to know are, given  a comma space x comma space y space greater than space 0:

    • log subscript a x y equals blank log subscript a x plus blank log subscript a y

      • This relates to a to the power of x cross times blank a to the power of y equals a to the power of x plus y end exponent

    • log subscript a x over y equals blank log subscript a x blank negative space log subscript a y 

      • This relates to a to the power of x divided by blank a to the power of y equals a to the power of x minus y end exponent

    • log subscript a x to the power of m equals blank m log subscript a x 

      • This relates to left parenthesis a to the power of x right parenthesis to the power of y equals a to the power of x y end exponent

  • These laws are in the formula booklet so you do not need to remember them

    • You must make sure you know how to use them

Laws of Logarithms Notes fig2

Useful results from the laws of logarithms

  • Given a space greater than space 0 space comma space a space not equal to space 1

    • log subscript a 1 equals blank 0

      • This is equivalent to a to the power of 0 equals 1

  • If we substitute b for a into the given identity in the formula booklet

    • a to the power of x equals b space left right double arrow space log subscript a b space equals space x where a space greater than space 0 comma space b space greater than space 0 comma space a space not equal to space 1

    • a to the power of x space equals space a space left right double arrow space log subscript a a space equals space x gives a to the power of 1 space equals space a space left right double arrow space log subscript a a space equals space 1 

      • This is an important and useful result

  • Substituting this into the third law gives the result

    • log subscript a a to the power of k equals blank k

  • Taking the inverse of its operation gives the result

    • a to the power of log subscript a x end exponent equals blank x

  • From the third law we can also conclude that

    • log subscript a 1 over x equals blank minus log subscript a x

Laws of Logarithms Notes fig3

  • These useful results are not in the formula booklet but can be deduced from the laws that are

  • Beware…

    • …log subscript a open parentheses x plus y close parentheses space not equal to space log subscript a x plus log subscript a y

  • These results apply to ln space x space left parenthesis log subscript e x right parenthesis too

    • Two particularly useful results are

      • ln space e to the power of x space equals space x

      • e to the power of ln x end exponent space equals space x

  • Laws of logarithms can be used to …

    • simplify expressions

    • solve logarithmic equations

    • solve exponential equations

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

a) Write the expression 2 space log space 4 space minus space log space 2 in the form log space k, where k space element of space straight integer numbers.

 

aa-sl-1-2-2-laws-of-logs-we-solution-part-a

b)   Hence, or otherwise, solve 2 space log space 4 minus log space 2 equals negative log blank 1 over x.

aa-sl-1-2-2-laws-of-logs-we-solution-part-b

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 yourselfFlashcards

    Did this page help you?

    Previous:ExponentsNext:Language of Sequences & Series
    Amber

    Author: Amber

    Expertise: Maths

    Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.