Logarithmic Functions (Edexcel IGCSE Further Pure Maths) : Revision Note

Written by: Amber

Reviewed by: Dan Finlay

Updated on 16 October 2024

Introduction to Logarithms

What are logarithms?

A logarithm is the inverse of an exponent
- If $a^{x} = b$ then $\log_{a} (b) = x$ , where $a > 0, b > 0, a \neq 1$
  - 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
- $\log_{a} (b) = x$ means “ $x$ is the power that you raise $a$ to, to get $b$ "
- So $\log_{5} 125 = 3$ means “3 is the power that you raise 5 to, to get 125”

Two important special cases are:
- $\ln x = \log_{e} (x)$
  - Where $e$ is the mathematical constant 2.718…
  - This is called the natural logarithm and will have its own button on your calculator
- $\log x = \log_{10} (x)$
  - Logarithms of base 10 are frequently used
  - They are often abbreviated simply as $\log x$

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 $2^{x} = 8$ we know that $x$ must be $3$
- But logarithms allow us to solve more complicated problems
  - For example, the equation $2^{x} = 10$ does not have an obvious answer
  - Instead we can rewrite the equation as a logarithm

$\log_{2} 10 = x$

and use our calculator to find the decimal value of $\log_{2} 10$

$x = 3.321928 . . .$

Examiner Tips and Tricks

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

Worked Example

Solve the following equations:

i) $x = \log_{3} 27$ ,
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"

$3^{x} = 27$

This can be solved by inspection

$3^{3} = 27$

$x = 3$

ii) $2^{x} = 21.4$ , 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"

$x = \log_{2} 21.4$

Use your calculator to find the value of that logarithm

$x = 4.419538 . . .$

Round to 3 significant figures

$x = 4.42$ (3 s.f.)

Logarithmic Functions & Graphs

What is a logarithmic function?

A logarithmic function is of the form $f (x) = \log_{a} x, x > 0$
- 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
$\log_{a} x$ and $a^{x}$ are inverse functions
- $\log_{a} (a^{x}) = x$ and $a^{\log_{a} x} = x$

What are the key features of logarithmic graphs?

The graph of $y = \log_{a} x$
- does not have a $y$ -intercept
- has a vertical asymptote at the y-axis: $x = 0$
- has one $x$ -intercept at (1, 0)
- passes through the point (a, 1)
- does not have any minimum or maximum points

Worked Example

On the same set of axes, sketch the graphs of $y = \log_{3} x$ and $y = 3^{x}$ . Be sure to label any axis intercepts.

$y = \log_{3} x$ will have the typical logarithmic graph shape
The $y$ -axis is an asymptote, and the $x$ -intercept is $(1, 0)$

$y = 3^{x}$ will have the typical exponential graph shape
The $x$ -axis is an asymptote, and the $y$ -intercept is $(0, 1)$

$\log_{3} x$ and $3^{x}$ are inverse functions
Therefore their graphs will be reflections of each other in the line $y = x$

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