IBMathsDPAnalysis & Approaches (AA)HLRevision Notes1. Number & AlgebraExponentials & LogsLaws of Logarithms

Laws of Logarithms (DP IB Analysis & Approaches (AA)): Revision Note

Last updated

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

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

Did this video help you?

Change of Base

Why change the base of a logarithm?

  • The laws of logarithms can only be used if the logs have the same base

    • If a problem involves logarithms with different bases, you can change the base of the logarithm and then apply the laws of logarithms

  • Changing the base of a logarithm can be particularly useful if you need to evaluate a log problem without a calculator

    • Choose the base such that you would know how to solve the problem from the equivalent exponent

How do I change the base of a logarithm?

  • The formula for changing the base of a logarithm is

begin mathsize 22px style log subscript a x equals blank fraction numerator log subscript b x over denominator log subscript b a end fraction end style

  • This is in the formula booklet

  • The value you choose for b does not matter, however if you do not have a calculator, you can choose b such that the problem will be possible to solve

Examiner Tips and Tricks

  • Changing the base is a key skill which can help you with many different types of questions, make sure you are confident with it

    • It is a particularly useful skill for examinations where a GDC is not permitted

Worked Example

By choosing a suitable value for b, use the change of base law to find the value of  log subscript 8 space end subscript 32 without using a calculator.

aa-sl-1-2-2-change-of-base-we-solution-1

You've read 1 of your 5 free revision notes this week

Unlock more, it's free!

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

the (exam) results speak for themselves:

  • I would just like to say a massive thank you for putting together such a brilliant, easy to use website.I really think using this site helped me secure my top gradesin science and maths. You really did save my exams! Thank you.

    Beth
    IGCSE Student

  • This website is soooo useful and I can’t ever thank you enough for organising questions by topic like this. Furthermore, the name of the website could not have been more appropriate as it literally did SAVE MY EXAMS!

    Fathima
    A Level Student
Test yourselfFlashcards

Did this page help you?

Previous:Introduction to LogarithmsNext:Solving Exponential Equations
Download notes on Laws of Logarithms