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DP IB Maths: AA SL

Revision Notes

Home / IB / Maths: AA SL / DP / Revision Notes / 1. Number & Algebra / 1.2 Exponentials & Logs / 1.2.2 Laws of Logarithms


1.2.2 Laws of Logarithms


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 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

Exam 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

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

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

Exam Tip

  • 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



  • 1. Number & Algebra
    • 1.1 Number Toolkit
      • 1.1.1 Standard Form
        • 1.1.2 Laws of Indices
        • 1.2 Exponentials & Logs
          • 1.2.1 Introduction to Logarithms
            • 1.2.2 Laws of Logarithms
              • 1.2.3 Solving Exponential Equations
              • 1.3 Sequences & Series
                • 1.3.1 Language of Sequences & Series
                  • 1.3.2 Arithmetic Sequences & Series
                    • 1.3.3 Geometric Sequences & Series
                      • 1.3.4 Applications of Sequences & Series
                        • 1.3.5 Compound Interest & Depreciation
                        • 1.4 Proof & Reasoning
                          • 1.4.1 Proof
                          • 1.5 Binomial Theorem
                            • 1.5.1 Binomial Theorem
                          • 2. Functions
                            • 2.1 Linear Functions & Graphs
                              • 2.1.1 Equations of a Straight Line
                              • 2.2 Quadratic Functions & Graphs
                                • 2.2.1 Quadratic Functions
                                  • 2.2.2 Factorising & Completing the Square
                                    • 2.2.3 Solving Quadratics
                                      • 2.2.4 Quadratic Inequalities
                                        • 2.2.5 Discriminants
                                        • 2.3 Functions Toolkit
                                          • 2.3.1 Language of Functions
                                            • 2.3.2 Composite & Inverse Functions
                                              • 2.3.3 Graphing Functions
                                              • 2.4 Further Functions & Graphs
                                                • 2.4.1 Reciprocal & Rational Functions
                                                  • 2.4.2 Exponential & Logarithmic Functions
                                                    • 2.4.3 Solving Equations
                                                      • 2.4.4 Modelling with Functions
                                                      • 2.5 Transformations of Graphs
                                                        • 2.5.1 Translations of Graphs
                                                          • 2.5.2 Reflections of Graphs
                                                            • 2.5.3 Stretches of Graphs
                                                              • 2.5.4 Composite Transformations of Graphs
                                                            • 3. Geometry & Trigonometry
                                                              • 3.1 Geometry Toolkit
                                                                • 3.1.1 Coordinate Geometry
                                                                  • 3.1.2 Radian Measure
                                                                    • 3.1.3 Arcs & Sectors
                                                                    • 3.2 Geometry of 3D Shapes
                                                                      • 3.2.1 3D Coordinate Geometry
                                                                        • 3.2.2 Volume & Surface Area
                                                                        • 3.3 Trigonometry
                                                                          • 3.3.1 Pythagoras & Right-Angled Triganometry
                                                                            • 3.3.2 Non Right-Angled Trigonometry
                                                                              • 3.3.3 Applications of Trigonometry & Pythagoras
                                                                              • 3.4 Further Trigonometry
                                                                                • 3.4.1 The Unit Circle
                                                                                  • 3.4.2 Exact Values
                                                                                  • 3.5 Trigonometric Functions & Graphs
                                                                                    • 3.5.1 Graphs of Trigonometric Functions
                                                                                      • 3.5.2 Transformations of Trigonometric Functions
                                                                                        • 3.5.3 Modelling with Trigonometric Functions
                                                                                        • 3.6 Trigonometric Equations & Identities
                                                                                          • 3.6.1 Simple Identities
                                                                                            • 3.6.2 Double Angle Formulae
                                                                                              • 3.6.3 Relationship Between Trigonometric Ratios
                                                                                                • 3.6.4 Linear Trigonometric Equations
                                                                                                  • 3.6.5 Quadratic Trigonometric Equations
                                                                                                • 4. Statistics & Probability
                                                                                                  • 4.1 Statistics Toolkit
                                                                                                    • 4.1.1 Sampling & Data Collection
                                                                                                      • 4.1.2 Statistical Measures
                                                                                                        • 4.1.3 Frequency Tables
                                                                                                          • 4.1.4 Linear Tranformations of Data
                                                                                                            • 4.1.5 Outliers
                                                                                                              • 4.1.6 Univariate Data
                                                                                                                • 4.1.7 Interpreting Data
                                                                                                                • 4.2 Correlation & Regression
                                                                                                                  • 4.2.1 Bivariate Data
                                                                                                                    • 4.2.2 Correlation & Regression
                                                                                                                    • 4.3 Probability
                                                                                                                      • 4.3.1 Probability & Types of Events
                                                                                                                        • 4.3.2 Conditional Probability
                                                                                                                          • 4.3.3 Sample Space Diagrams
                                                                                                                          • 4.4 Probability Distributions
                                                                                                                            • 4.4.1 Discrete Probability Distributions
                                                                                                                              • 4.4.2 Expected Values
                                                                                                                              • 4.5 Binomial Distribution
                                                                                                                                • 4.5.1 The Binomial Distribution
                                                                                                                                  • 4.5.2 Calculating Binomial Probabilities
                                                                                                                                  • 4.6 Normal Distribution
                                                                                                                                    • 4.6.1 The Normal Distribution
                                                                                                                                      • 4.6.2 Calculations with Normal Distribution
                                                                                                                                        • 4.6.3 Standardisation of Normal Variables
                                                                                                                                      • 5. Calculus
                                                                                                                                        • 5.1 Differentiation
                                                                                                                                          • 5.1.1 Introduction to Differentiation
                                                                                                                                            • 5.1.2 Applications of Differentiation
                                                                                                                                            • 5.2 Further Differentiation
                                                                                                                                              • 5.2.1 Differentiating Special Functions
                                                                                                                                                • 5.2.2 Techniques of Differentiation
                                                                                                                                                  • 5.2.3 Second Order Derivatives
                                                                                                                                                    • 5.2.4 Further Applications of Differentiation
                                                                                                                                                      • 5.2.5 Concavity & Points of Inflection
                                                                                                                                                        • 5.2.6 Derivatives & Graphs
                                                                                                                                                        • 5.3 Integration
                                                                                                                                                          • 5.3.1 Introduction to Integration
                                                                                                                                                            • 5.3.2 Applications of Integration
                                                                                                                                                            • 5.4 Further Integration
                                                                                                                                                              • 5.4.1 Integrating Special Functions
                                                                                                                                                                • 5.4.2 Techniques of Integration
                                                                                                                                                                  • 5.4.3 Definite Integrals
                                                                                                                                                                    • 5.4.4 Further Applications of Integration
                                                                                                                                                                    • 5.5 Optimisation
                                                                                                                                                                      • 5.5.1 Modelling with Differentiation
                                                                                                                                                                      • 5.6 Kinematics
                                                                                                                                                                        • 5.6.1 Kinematics Toolkit
                                                                                                                                                                          • 5.6.2 Calculus for Kinematics
                                                                                                                                                                        Amber Perry

                                                                                                                                                                        Author: Amber

                                                                                                                                                                        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.


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