Laws of Logarithms

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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 $a, x, y > 0$ :
- $\log_{a} x y = \log_{a} x + \log_{a} y$
  - This relates to $a^{x} \times a^{y} = a^{x + y}$
- $\log_{a} \frac{x}{y} = \log_{a} x {- \log}_{a} y$
  - This relates to $a^{x} \div a^{y} = a^{x - y}$
- $\log_{a} x^{m} = m \log_{a} x$
  - This relates to $(a^{x})^{y} = a^{x y}$
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 $a > 0, a \neq 1$
- $\log_{a} 1 = 0$
  - This is equivalent to $a^{0} = 1$
If we substitute b for a into the given identity in the formula booklet
- $a^{x} = b \Leftrightarrow \log_{a} b = x$ where $a > 0, b > 0, a \neq 1$
- $a^{x} = a \Leftrightarrow \log_{a} a = x$ gives $a^{1} = a \Leftrightarrow \log_{a} a = 1$
  - This is an important and useful result
Substituting this into the third law gives the result
- $\log_{a} a^{k} = k$
Taking the inverse of its operation gives the result
- $a^{\log_{a} x} = x$
From the third law we can also conclude that
- $\log_{a} \frac{1}{x} = - \log_{a} x$

Laws of Logarithms Notes fig3

These useful results are not in the formula booklet but can be deduced from the laws that are
Beware…
- … $\log_{a} (x + y) \neq \log_{a} x + \log_{a} y$
These results apply to $\ln x (\log_{e} x)$ too
- Two particularly useful results are
  - $\ln e^{x} = x$
  - $e^{\ln x} = x$
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

Write the expression

2 \log 4 - \log 2

in the form

\log k

, where

k \in ℤ

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

b) Hence, or otherwise, solve

2 \log 4 - \log 2 = - \log \frac{1}{x}

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

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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

$\log_{a} x = \frac{\log_{b} x}{\log_{b} a}$

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 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 $\log_{8} 32$ without using a calculator.

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

Laws of Logarithms (DP IB Maths: AA HL)

Revision Note

What are the laws of logarithms?

Useful results from the laws of logarithms

Why change the base of a logarithm?

How do I change the base of a logarithm?

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1. Number & Algebra

1.1 Number & Algebra Toolkit

1.1.1 Standard Form

1.1.2 Laws of Indices

1.1.3 Partial Fractions

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 Simple Proof & Reasoning

1.4.1 Proof

1.5 Further Proof & Reasoning

1.5.1 Proof by Induction

1.5.2 Proof by Contradiction

1.6 Binomial Theorem

1.6.1 Binomial Theorem

1.6.2 Extension of The Binomial Theorem

1.7 Permutations & Combinations

1.7.1 Counting Principles

1.7.2 Permutations & Combinations

1.8 Complex Numbers

1.8.1 Intro to Complex Numbers

1.8.2 Modulus & Argument

1.8.3 Introduction to Argand Diagrams

1.9 Further Complex Numbers

1.9.1 Geometry of Complex Numbers

1.9.2 Forms of Complex Numbers

1.9.3 Complex Roots of Polynomials

1.9.4 De Moivre's Theorem

1.9.5 Roots of Complex Numbers

1.10 Systems of Linear Equations

1.10.1 Systems of Linear Equations

1.10.2 Algebraic Solutions

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 Symmetry of Functions

2.3.4 Graphing Functions

2.4 Other Functions & Graphs

2.4.1 Exponential & Logarithmic Functions

2.4.2 Solving Equations

2.4.3 Modelling with Functions

2.5 Reciprocal & Rational Functions

2.5.1 Reciprocal & Rational Functions

2.6 Transformations of Graphs

2.6.1 Translations of Graphs

2.6.2 Reflections of Graphs

2.6.3 Stretches Graphs

2.6.4 Composite Transformations of Graphs

2.7 Polynomial Functions

2.7.1 Factor & Remainder Theorem

2.7.2 Polynomial Division

2.7.3 Polynomial Functions

2.7.4 Roots of Polynomials

2.8 Inequalities

2.8.1 Solving Inequalities Graphically