Graphs of logarithmic functions
For any given base, $b$b, the graph of a logarithmic function $y=\log_bx$y=logbx is related to the graph of the exponential function $y=b^x$y=bx. In particular, they are a reflection of each other across the line $y=x$y=x. This is because exponential and logarithmic functions are inverse functions. It may be useful to re-visit exponential graphs and their transformations before working through this section.
Graphs of $b^x$bx and $\log_bx$logbx, for $b>1$b>1
As $b^x$bx and $\log_bx$logbx are a reflection of each other, we can observe the following properties of the graphs of logarithmic functions of the form $y=\log_bx$y=logbx:
- Domain: the argument, $x$x, is restricted to only positive values. That is, $x>0$x>0.
- Range: all real $y$y values can be obtained.
- Asymptotes: there is a vertical asymptote at $x=0$x=0 (on the $y$y-axis) for any logarithmic function of the form $y=\log_bx$y=logbx, regardless of the base. As a result, there is no $y$y-intercept.
- $x$x-intercept: The logarithm of $1$1 is $0$0, irrespective of the base used. As a result, the graph of $y=\log_bx$y=logbx intersects the $x$x-axis at $\left(1,0\right)$(1,0).
Two particular log curves from the family of log functions with $b>1$b>1 are shown below. The top curve is the graph of the function $f\left(x\right)=\log_2\left(x\right)$f(x)=log2(x), and the bottom curve is the graph of the function $g\left(x\right)=\log_4\left(x\right)$g(x)=log4(x), as labelled.
The points shown on each curve help to demonstrate the way the gradient of the curve changes as $b$b increases in value. The larger the value of the base the less steep the graph is. Each graph will pass through $\left(b,1\right)$(b,1), since the $\log_bb=1$logbb=1. Thus, the higher base graph will appear closer to the $x$x-axis.
Just as for exponential functions if the base $b$b is greater than $1$1 then the function increases across the entire domain and for $0
Practice question
Question 1
Consider the function $y=\log_4x$y=log4x, the graph of which has been sketched below.
Complete the following table of values.
$x$x $\frac{1}{16}$116 $\frac{1}{4}$14 $4$4 $16$16 $256$256 $y$y $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Determine the $x$x-value of the $x$x-intercept of $y=\log_4x$y=log4x.
How many $y$y-intercepts does $\log_4x$log4x have?
Determine the $x$x value for which $\log_4x=1$log4x=1.
Question 2
The functions $y=\log_2x$y=log2x and $y=\log_3x$y=log3x have been graphed on the same set of axes.
Using the labels $A$A and $B$B, state which graph corresponds to each function.
$y=\log_2x$y=log2x is labelled $\editable{}$ and $y=\log_3x$y=log3x is labelled $\editable{}$.
Which of the following multiple choice answers completes the statement?
"The larger the base $a$a of the function $y=\log_ax$y=logax, ..."
...the further the asymptote is from the $y$y-axis.
A...the less steep the graph.
B...the steeper the graph.
C...the further the $x$x-intercept is from the $y$y-axis
D