4.3 Graphs of logarithmic functions  (Page 7/8)

Finding the equation from a graph

Find a possible equation for the common logarithmic function graphed in [link] .

This graph has a vertical asymptote at $x=\mathrm{–2}$ and has been vertically reflected. We do not know yet the vertical shift or the vertical stretch. We know so far that the equation will have form:

$$f(x)=-a\log (x+2)+k$$

It appears the graph passes through the points $\left(\mathrm{–1},1\right)$ and $\left(2,\mathrm{–1}\right).$ Substituting $\left(\mathrm{–1},1\right),$

$$\begin{array}{ll}1=-a\log (\mathrm{-1}+2)+k\hfill & \text{Substitute }(\mathrm{-1},1).\hfill \\ 1=-a\log (1)+k\hfill & \text{Arithmetic}.\hfill \\ 1=k\hfill & \text{log(1)}=0.\hfill \end{array}$$

Next, substituting in $\left(2,\mathrm{–1}\right)$ ,

$$\begin{array}{lll}-1=-a\log (2+2)+1\hfill & \hfill & \text{Plug in }(2,\mathrm{-1}).\hfill \\ -2=-a\log (4)\hfill & \hfill & \text{Arithmetic}.\hfill \\ a=\frac{2}{\log (4)}\hfill & \hfill & \text{Solve for }a.\hfill \end{array}$$

This gives us the equation $f(x)=–\frac{2}{\log (4)}\log (x+2)+1.$

Try It

Give the equation of the natural logarithm graphed in [link] .

$f(x)=2\ln (x+3)-1$

Key equations

Key concepts

• To find the domain of a logarithmic function, set up an inequality showing the argument greater than zero, and solve for $x.$ See [link] and [link]
• The graph of the parent function $f(x)={\log }_b\left(x\right)$ has an x- intercept at $\left(1,0\right),$ domain $\left(0,\infty \right),$ range $\left(-\infty ,\infty \right),$ vertical asymptote $x=0,$ and
  • if $b>1,$ the function is increasing.
  • if $0<b<1,$ the function is decreasing.
  See [link] .
• The equation $f(x)={\log }_b\left(x+c\right)$ shifts the parent function $y={\log }_b\left(x\right)$ horizontally
  • left $c$ units if $c>0.$
  • right $c$ units if $c<0.$
  See [link] .
• The equation $f(x)={\log }_b\left(x\right)+d$ shifts the parent function $y={\log }_b\left(x\right)$ vertically
  • up $d$ units if $d>0.$
  • down $d$ units if $d<0.$
  See [link] .
• For any constant $a>0,$ the equation $f(x)=a{\log }_b\left(x\right)$
  • stretches the parent function $y={\log }_b\left(x\right)$ vertically by a factor of $a$ if $\left|a\right|>1.$
  • compresses the parent function $y={\log }_b\left(x\right)$ vertically by a factor of $a$ if $\left|a\right|<1.$
  See [link] and [link] .
• When the parent function $y={\log }_b\left(x\right)$ is multiplied by $-1,$ the result is a reflection about the x -axis. When the input is multiplied by $-1,$ the result is a reflection about the y -axis.
  • The equation $f(x)=-{\log }_b\left(x\right)$ represents a reflection of the parent function about the x- axis.
  • The equation $f(x)={\log }_b\left(-x\right)$ represents a reflection of the parent function about the y- axis.
  See [link] .
  • A graphing calculator may be used to approximate solutions to some logarithmic equations See [link] .
• All translations of the logarithmic function can be summarized by the general equation $f(x)=a{\log }_b\left(x+c\right)+d.$ See [link] .
• Given an equation with the general form $f(x)=a{\log }_b\left(x+c\right)+d,$ we can identify the vertical asymptote $x=-c$ for the transformation. See [link] .
• Using the general equation $f(x)=a{\log }_b\left(x+c\right)+d,$ we can write the equation of a logarithmic function given its graph. See [link] .