# 6.4 Graphs of logarithmic functions  (Page 7/8)

 Page 7 / 8
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] .

what is the coefficient of -4×
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
An investment account was opened with an initial deposit of \$9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
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Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
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Abhi
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Commplementary angles
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Nharnhar
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a perfect square v²+2v+_
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