6.4 Graphs of logarithmic functions  (Page 7/8)

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Translations of the Parent Function $\text{\hspace{0.17em}}y={\mathrm{log}}_{b}\left(x\right)$
Translation Form
Shift
• Horizontally $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ units to the left
• Vertically $\text{\hspace{0.17em}}d\text{\hspace{0.17em}}$ units up
$y={\mathrm{log}}_{b}\left(x+c\right)+d$
Stretch and Compress
• Stretch if $\text{\hspace{0.17em}}|a|>1$
• Compression if $\text{\hspace{0.17em}}|a|<1$
$y=a{\mathrm{log}}_{b}\left(x\right)$
Reflect about the x -axis $y=-{\mathrm{log}}_{b}\left(x\right)$
Reflect about the y -axis $y={\mathrm{log}}_{b}\left(-x\right)$
General equation for all translations $y=a{\mathrm{log}}_{b}\left(x+c\right)+d$

Translations of logarithmic functions

All translations of the parent logarithmic function, $\text{\hspace{0.17em}}y={\mathrm{log}}_{b}\left(x\right),$ have the form

where the parent function, $\text{\hspace{0.17em}}y={\mathrm{log}}_{b}\left(x\right),b>1,$ is

• shifted vertically up $\text{\hspace{0.17em}}d\text{\hspace{0.17em}}$ units.
• shifted horizontally to the left $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ units.
• stretched vertically by a factor of $\text{\hspace{0.17em}}|a|\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}|a|>0.$
• compressed vertically by a factor of $\text{\hspace{0.17em}}|a|\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}0<|a|<1.$
• reflected about the x- axis when $\text{\hspace{0.17em}}a<0.$

For $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{log}\left(-x\right),$ the graph of the parent function is reflected about the y -axis.

Finding the vertical asymptote of a logarithm graph

What is the vertical asymptote of $\text{\hspace{0.17em}}f\left(x\right)=-2{\mathrm{log}}_{3}\left(x+4\right)+5?$

The vertical asymptote is at $\text{\hspace{0.17em}}x=-4.$

What is the vertical asymptote of $\text{\hspace{0.17em}}f\left(x\right)=3+\mathrm{ln}\left(x-1\right)?$

$x=1$

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 $\text{\hspace{0.17em}}x=–2\text{\hspace{0.17em}}$ 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:

$f\left(x\right)=-a\mathrm{log}\left(x+2\right)+k$

It appears the graph passes through the points $\text{\hspace{0.17em}}\left(–1,1\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(2,–1\right).\text{\hspace{0.17em}}$ Substituting $\text{\hspace{0.17em}}\left(–1,1\right),$

Next, substituting in $\text{\hspace{0.17em}}\left(2,–1\right)$ ,

This gives us the equation $\text{\hspace{0.17em}}f\left(x\right)=–\frac{2}{\mathrm{log}\left(4\right)}\mathrm{log}\left(x+2\right)+1.$

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

$f\left(x\right)=2\mathrm{ln}\left(x+3\right)-1$

Is it possible to tell the domain and range and describe the end behavior of a function just by looking at the graph?

Yes, if we know the function is a general logarithmic function. For example, look at the graph in [link] . The graph approaches $\text{\hspace{0.17em}}x=-3\text{\hspace{0.17em}}$ (or thereabouts) more and more closely, so $\text{\hspace{0.17em}}x=-3\text{\hspace{0.17em}}$ is, or is very close to, the vertical asymptote. It approaches from the right, so the domain is all points to the right, $\text{\hspace{0.17em}}\left\{x\text{\hspace{0.17em}}|\text{\hspace{0.17em}}x>-3\right\}.\text{\hspace{0.17em}}$ The range, as with all general logarithmic functions, is all real numbers. And we can see the end behavior because the graph goes down as it goes left and up as it goes right. The end behavior is that as $\text{\hspace{0.17em}}x\to -{3}^{+},f\left(x\right)\to -\infty \text{\hspace{0.17em}}$ and as $\text{\hspace{0.17em}}x\to \infty ,f\left(x\right)\to \infty .$

Access these online resources for additional instruction and practice with graphing logarithms.

Key equations

 General Form for the Translation of the Parent Logarithmic Function

Key concepts

• To find the domain of a logarithmic function, set up an inequality showing the argument greater than zero, and solve for $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ See [link] and [link]
• The graph of the parent function $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ has an x- intercept at $\text{\hspace{0.17em}}\left(1,0\right),$ domain $\text{\hspace{0.17em}}\left(0,\infty \right),$ range $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ vertical asymptote $\text{\hspace{0.17em}}x=0,$ and
• if $\text{\hspace{0.17em}}b>1,$ the function is increasing.
• if $\text{\hspace{0.17em}}0 the function is decreasing.
• The equation $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x+c\right)\text{\hspace{0.17em}}$ shifts the parent function $\text{\hspace{0.17em}}y={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ horizontally
• left $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ units if $\text{\hspace{0.17em}}c>0.$
• right $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ units if $\text{\hspace{0.17em}}c<0.$
• The equation $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x\right)+d\text{\hspace{0.17em}}$ shifts the parent function $\text{\hspace{0.17em}}y={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ vertically
• up $\text{\hspace{0.17em}}d\text{\hspace{0.17em}}$ units if $\text{\hspace{0.17em}}d>0.$
• down $\text{\hspace{0.17em}}d\text{\hspace{0.17em}}$ units if $\text{\hspace{0.17em}}d<0.$
• For any constant $\text{\hspace{0.17em}}a>0,$ the equation $\text{\hspace{0.17em}}f\left(x\right)=a{\mathrm{log}}_{b}\left(x\right)$
• stretches the parent function $\text{\hspace{0.17em}}y={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ vertically by a factor of $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}|a|>1.$
• compresses the parent function $\text{\hspace{0.17em}}y={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ vertically by a factor of $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}|a|<1.$
• When the parent function $\text{\hspace{0.17em}}y={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ is multiplied by $\text{\hspace{0.17em}}-1,$ the result is a reflection about the x -axis. When the input is multiplied by $\text{\hspace{0.17em}}-1,$ the result is a reflection about the y -axis.
• The equation $\text{\hspace{0.17em}}f\left(x\right)=-{\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ represents a reflection of the parent function about the x- axis.
• The equation $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(-x\right)\text{\hspace{0.17em}}$ represents a reflection of the parent function about the y- axis.
• 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 See [link] .
• Given an equation with the general form we can identify the vertical asymptote $\text{\hspace{0.17em}}x=-c\text{\hspace{0.17em}}$ for the transformation. See [link] .
• Using the general equation $\text{\hspace{0.17em}}f\left(x\right)=a{\mathrm{log}}_{b}\left(x+c\right)+d,$ we can write the equation of a logarithmic function given its graph. See [link] .

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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