You need to have JavaScript enabled in order to access this site.

Graphing Translated Logarithmic Functions

Examples

Graph each function. Give the domain and range.

$LaTeX: f\left(x\right)=\log_2\left(x-1\right)$ $f\left(x\right)=\log_2\left(x-1\right)$
$LaTeX: f\left(x\right)=\left(\log_3x\right)-1$ $f\left(x\right)=\left(\log_3x\right)-1$
$LaTeX: f\left(x\right)=\log_4\left(x+2\right)+1$ $f\left(x\right)=\log_4\left(x+2\right)+1$

Solutions

$LaTeX: f\left(x\right)=\log_2\left(x-1\right)$ $f\left(x\right)=\log_2\left(x-1\right)$
The graph of $LaTeX: f\left(x\right)=\log_2\left(x-1\right)$ $f\left(x\right)=\log_2\left(x-1\right)$ is the graph of $LaTeX: f\left(x\right)=\log_2x\:$ $f\left(x\right)=\log_2x\:$ translated 1 unit to the right. The vertical asymptote has equation $LaTeX: x=1.\:$ $x=1.\:$ Since logarithms can be found only for positive numbers, we solve $LaTeX: x-1>0\:$ $x-1>0\:$ to find the domain, $LaTeX: \left(1,\infty\right).\:$ $\left(1,\infty\right).\:$ To determine ordered pairs to plot, use the equivalent exponential form of the equation $LaTeX: f\left(x\right)=\log_2\left(x-1\right)$ $f\left(x\right)=\log_2\left(x-1\right)$ .
$LaTeX: y=\log_x\left(x-1\right)$ $y=\log_x\left(x-1\right)$
$x-1=2^y$
$x=2^y+1$

We first choose values for y and then calculate each of the corresponding
x-values. The range is $LaTeX: \left(-\infty,\infty\right).$ $\left(-\infty,\infty\right).$
$LaTeX: f\left(x\right)=\left(\log_3x\right)-1$ $f\left(x\right)=\left(\log_3x\right)-1$
The function $LaTeX: f\left(x\right)=\left(\log_3x\right)-1$ $f\left(x\right)=\left(\log_3x\right)-1$ has the same graph as $LaTeX: g\left(x\right)=\log_3x\:$ $g\left(x\right)=\log_3x\:$ translated 1 unit down. We find the ordered pairs to plot by writing the equation $LaTeX: f\left(x\right)=\left(\log_3x\right)-1$ $f\left(x\right)=\left(\log_3x\right)-1$ in exponential form.
$LaTeX: f\left(x\right)=\left(\log_3x\right)-1$ $f\left(x\right)=\left(\log_3x\right)-1$
$LaTeX: y+1=\log_3x$ $y+1=\log_3x$
$LaTeX: x=3^{x+1}$ $x=3^{x+1}$

Again, choose y-values and calculate the corresponding
x-values. The domain is $LaTeX: \left(0,\infty\right)\:$ $\left(0,\infty\right)\:$ and the range is $LaTeX: \left(-\infty,\infty\right)$ $\left(-\infty,\infty\right)$ .
$LaTeX: f\left(x\right)=\log_4\left(x+2\right)+1$ $f\left(x\right)=\log_4\left(x+2\right)+1$
The graph of $LaTeX: f\left(x\right)=\log_4\left(x+2\right)+1$ $f\left(x\right)=\log_4\left(x+2\right)+1$ is obtained by shifting the graph of $LaTeX: y=\log_3x\:$ $y=\log_3x\:$ to the left 2 units and up 1 unit. The domain is found by solving $x+2>0$ , which yields $LaTeX: \left(-2,\infty\right)$ $\left(-2,\infty\right)$ . The vertical asymptote has been shifted to the left 2 units as well, and it has equation $x=-2.$ The range is unaffected by the vertical shift and remains $LaTeX: \left(-\infty,\infty\right)$ $\left(-\infty,\infty\right)$ .

Example

This video shows an example of how to recognize the equation for a graph of a logarithmic function. It is the same process as the examples above, but going in the reverse direction. Please watch the Khan video below.