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

Introduction to Logarithms

What are logarithms?

• A logarithm is the inverse of an exponent

  • If  $a^x=b$  then  ${\log }_a\left(b\right)=x$,  where  $a>0,b>0,a\ne 1$

    • 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

  • ${\log }_a\left(b\right)=x$  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:

  • $\ln x={\log }_{\mathrm{e}}\left(x\right)$

    • Where $\mathrm{e}$ is the mathematical constant 2.718…

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

  • $\log x={\log }_{10}\left(x\right)$

    • Logarithms of base 10 are frequently used

    • They are often abbreviated simply as $\mathbf{log}\mathit{x}$

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 ${\log }_{2}10$

$x=3.321928...$

Logarithmic Functions & Graphs

What is a logarithmic function?

• A logarithmic function is of the form $\mathrm{f}\left(x\right)={\log }_{a}x,x>0$

  • 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

  • ${\log }_a\left(a^x\right)=x$  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 $\mathit{y}$-intercept

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

  • has one $\mathit{x}$-intercept at (1, 0)

  • passes through the point (a, 1)

  • does not have any minimum or maximum points