4.4 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] .

give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this
what is functions?
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
can you not take the square root of a negative number
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
can I get some pretty basic questions
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
what is the domain of f(x)=x-4/x^2-2x-15 then
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena