Logarithmic Functions (Edexcel IGCSE Further Pure Maths)

Revision Note

Amber

Written by: Amber

Reviewed by: Dan Finlay

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 greater than 0 comma space b greater than 0 comma space a not equal to 1

      • This is the essential definition of a logarithm

      • The number a is called the base of the logarithm

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

    • log subscript a left parenthesis b right parenthesis space equals space x  means “x is the power that you raise a to, to get b"

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

  • Two important special cases are:

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

      • Where straight e is the mathematical constant 2.718…

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

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

      • Logarithms of base 10 are frequently used

      • They are often abbreviated simply as bold log bold italic 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  2 to the power of x equals 8  we know that x must be 3

    • But logarithms allow us to solve more complicated problems

      • For example, the equation  2 to the power of x equals 10  does not have an obvious answer

      • Instead we can rewrite the equation as a logarithm

log subscript 2 10 equals x

  • and use our calculator to find the decimal value of log subscript 2 10

x equals 3.321928...

Examiner Tips and Tricks

  • Make sure you are completely familiar with your calculator's logarithm functions

Worked Example

Solve the following equations:

i) x equals log subscript 3 27,
Use the definition of a logarithm to rewrite this as an exponential equation

Remember, this equation means "x is the power that you raise 3 to, to get 27"


3 to the power of x equals 27


This can be solved by inspection

3 cubed equals 27

bold italic x bold equals bold 3

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


Use the definition of a logarithm to rewrite this as a logarithm equation

We want the logarithm that means "x is the power that you raise 2 to, to get 21.4"


x equals log subscript 2 21.4


Use your calculator to find the value of that logarithm


x equals 4.419538...


Round to 3 significant figures

bold italic x bold equals bold 4 bold. bold 42 (3 s.f.)

Logarithmic Functions & Graphs

What is a logarithmic function?

  • A logarithmic function is of the form space straight f left parenthesis x right parenthesis equals log subscript a x comma space x greater than 0

    • In this course the base a for a logarithmic function will always be an integer greater than one

  • Its domain is the set of all positive real numbers

    • You can't take a log of zero or a negative number

  • Its range is the set of all real numbers

  • log subscript a x and a to the power of x are inverse functions

    • log subscript a open parentheses a to the power of x close parentheses equals x  and  a to the power of log subscript a x end exponent equals x

What are the key features of logarithmic graphs?

  • The graph of  y equals log subscript a x

    • does not have a bold italic y-intercept

    • has a vertical asymptote at the y-axis: x equals 0

    • has one bold italic x-intercept at (1, 0)

    • passes through the point (a, 1)

    • does not have any minimum or maximum points

FsosVIe~_logarithm-graph

Worked Example

On the same set of axes, sketch the graphs of  y equals log subscript 3 x  and  y equals 3 to the power of x.  Be sure to label any axis intercepts.

y equals log subscript 3 x will have the typical logarithmic graph shape
The y-axis is an asymptote, and the x-intercept is open parentheses 1 comma 0 close parentheses

y equals 3 to the power of x will have the typical exponential graph shape
The x-axis is an asymptote, and the y-intercept is open parentheses 0 comma space 1 close parentheses

log subscript 3 x and 3 to the power of xare inverse functions
Therefore their graphs will be reflections of each other in the line y equals x

Graph of y=log_3(x) and  y=3^x

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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.

Dan Finlay

Author: Dan Finlay

Expertise: Maths Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.