Revision Notes Exam Questions Past Papers

A LevelMaths: Pure 3CIERevision Notes2. Logs & Exponentials2.2 Laws of Logarithms2.2.1 Laws of Logarithms

Laws of Logarithms (CIE A Level Maths: Pure 3)

Revision Note

Author

Paul

Last updated

19 April 2023

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
- a^m x aⁿ = a^m+n
- (a^m)ⁿ = a^mn
There are equivalent laws of logarithms (for a > 0)
- $\log_{a} x y = \log_{a} x + \log_{a} y$
- $\log_{a} (\frac{x}{y}) = \log_{a} x - \log_{a} y$
- $\log_{a} x^{k} = k \log_{a} x$

Laws of Logarithms Notes fig2, A Level & AS Level Pure Maths Revision Notes

There are also some particular results these lead to
- $\log_{a} a = 1$
- $\log_{a} a^{x} = x$
- $a^{\log_{a} x} = x$
- $\log_{a} 1 = 0$
- $\log_{a} (\frac{1}{x}) = - \log_{a} x$

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
- $\ln x \equiv \log_{e} x$
- In particular $e^{\ln x} = x$ and $\ln (e^{x}) = x$

How do I use the laws of logarithms?

Laws of logarithms can be used to …
- … simplify expressions
- … solve logarithmic equations
- … solve exponential equations

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

2-2-1-laws-of-logs-we-solution-1

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:2.1.4 Derivatives of Exponential FunctionsNext:2.2.2 Exponential Equations

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.

Laws of Logarithms (CIE A Level Maths: Pure 3)

Revision Note

What are the laws of logarithms?

How do I use the laws of logarithms?

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

1. Algebra & Functions

1.1 Modulus Functions

1.1.1 Modulus Functions - Sketching Graphs

1.1.2 Modulus Functions - Solving Equations

1.2 Polynomials

1.2.1 Polynomial Division

1.2.2 Factor & Remainder Theorem

1.2.3 Factorisation

1.2.4 Rational Expressions

1.2.5 Top Heavy Rational Expressions

1.3 Partial Fractions

1.3.1 Linear Denominators

1.3.2 Squared Linear Denominators

1.3.3 Quadratic Denominators

1.4 Further Modelling with Functions

1.4.1 Further Modelling with Functions

1.5 General Binomial Expansion

1.5.1 General Binomial Expansion

1.5.2 General Binomial Expansion - Subtleties

1.5.3 General Binomial Expansion - Multiple

1.5.4 Approximating values

2. Logs & Exponentials

2.1 Logarithmic & Exponential Function

2.1.1 Exponential Functions

2.1.2 Logarithmic Functions

2.1.3 "e"

2.1.4 Derivatives of Exponential Functions

2.2 Laws of Logarithms

2.2.1 Laws of Logarithms

2.2.2 Exponential Equations

2.3 Modelling with Logs & Exponentials

2.3.1 Exponential Growth & Decay

2.3.2 Using Exps & Logs in Modelling

2.3.3 Using Log Graphs in Modelling

3. Trigonometry

3.1 Reciprocal Trigonometric Functions

3.1.1 Reciprocal Trig Functions - Definitions

3.1.2 Reciprocal Trig Functions - Graphs

3.1.3 Trigonometry - Further Identities

3.2 Compound & Double Angle Formulae

3.2.1 Compound Angle Formulae

3.2.2 Double Angle Formulae

3.2.3 R addition formulae Rcos Rsin etc

3.3 Further Trigonometric Equations

3.3.1 Strategy for Further Trigonometric Equations

3.4 Trigonometric Proof

3.4.1 Trigonometric Proof

4. Differentiation

4.1 Further Differentiation

4.1.1 Differentiating Other Functions (Trig, ln & e etc)

4.1.2 Product Rule

4.1.3 Quotient Rule

4.2 Implicit Differentiation

4.2.1 Implicit Differentiation

4.3 Differentiation of Parametric Equations

4.3.1 Parametric Equations - Basics

4.3.2 Parametric Equations - Eliminating the Parameter

4.3.3 Parametric Equations - Sketching Graphs

4.3.4 Parametric Differentiation

5. Integration

5.1 Further Integration

5.1.1 Integrating Other Functions (Trig, ln & e etc)

5.1.2 Integrating with Trigonometric Identities

5.1.3 f'(x)/f(x)

5.1.4 Substitution (Reverse Chain Rule)

5.1.5 Harder Substitution

5.1.6 Integration by Parts

5.1.7 Integration using Partial Fractions

5.1.8 Integration Decision Making

5.2 Differential Equations

5.2.1 General Solutions

5.2.2 Particular Solutions

5.2.3 Separation of Variables