Logarithmic Functions (Edexcel IGCSE Further Maths)

Introduction to Logarithms

Spec point — 1A

What are logarithms?

• A logarithm is the inverse of an exponent

  • If $a^{x}=b$ then log_ a (b)=x, where $a>0,b>0,a\neq1$

    • 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

  • $\mathrm{log}_{a}(b)=x$ means “$x$ is the power that you raise $a$ to, to get $b$"
  • So $\mathrm{log}_{5}125=3$ means “3 is the power that you raise 5 to, to get 125”

• Two important special cases are:

  • $\ln x=log_ \mathrm{e} (x)$

    • 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 $\mathrm{log}x$

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

$\mathrm{log}_{2}10=x$

• and use our calculator to find the decimal value of $\mathrm{log}_{2}10$

$x=3.321928...$

Exam Hint

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

Worked Example Solve the following equations:

i) $x=\mathrm{log}_{3}27$, 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"

$x=\mathrm{log}_{2}21.4$

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

Spec point — 4A

What is a logarithmic function?

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

  • 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

• $\mathrm{log}_{a}x$ and $a^{x}$ are inverse functions

  • log_ a (a^x)=x and $a^{\mathrm{log}_{a}x}=x$

What are the key features of logarithmic graphs?

• The graph of $y=\mathrm{log}_{a}x$

  • 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 $y=\mathrm{log}_{3}x$ and $y=3^{x}$. Be sure to label any axis intercepts.

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

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

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