Logarithmic Functions (Edexcel IGCSE Further Pure Maths)

Introduction to Logarithms

What are logarithms?

• A logarithm is the inverse of an exponent

  • If    then  ,  where 

    • This is the essential definition of a logarithm

    • The number is called the base of the logarithm

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

  •   means “ is the power that you raise to, to get "

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

• Two important special cases are:

  • 

    • Where is the mathematical constant 2.718…

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

  • 

    • Logarithms of base 10 are frequently used

    • They are often abbreviated simply as

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    we know that must be

  • But logarithms allow us to solve more complicated problems

    • For example, the equation    does not have an obvious answer

    • Instead we can rewrite the equation as a logarithm

• and use our calculator to find the decimal value of

Logarithmic Functions & Graphs

What is a logarithmic function?

• A logarithmic function is of the form 

  • In this course the base 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

• and  are inverse functions

  •   and  

What are the key features of logarithmic graphs?

• The graph of 

  • does not have a -intercept

  • has a vertical asymptote at the y-axis: 

  • has one -intercept at (1, 0)

  • passes through the point (a, 1)

  • does not have any minimum or maximum points