# 6.4 Graphs of logarithmic functions  (Page 5/8)

 Page 5 / 8

Given a logarithmic function with the form $\text{\hspace{0.17em}}f\left(x\right)=a{\mathrm{log}}_{b}\left(x\right),$ $a>0,$ graph the translation.

1. Identify the vertical stretch or compressions:
• If $\text{\hspace{0.17em}}|a|>1,$ the graph of $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ is stretched by a factor of $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ units.
• If $\text{\hspace{0.17em}}|a|<1,$ the graph of $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ is compressed by a factor of $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ units.
2. Draw the vertical asymptote $\text{\hspace{0.17em}}x=0.$
3. Identify three key points from the parent function. Find new coordinates for the shifted functions by multiplying the $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ coordinates by $\text{\hspace{0.17em}}a.$
4. Label the three points.
5. The domain is $\text{\hspace{0.17em}}\left(0,\infty \right),$ the range is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ and the vertical asymptote is $\text{\hspace{0.17em}}x=0.$

## Graphing a stretch or compression of the parent function y = log b ( x )

Sketch a graph of $\text{\hspace{0.17em}}f\left(x\right)=2{\mathrm{log}}_{4}\left(x\right)\text{\hspace{0.17em}}$ alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

Since the function is $\text{\hspace{0.17em}}f\left(x\right)=2{\mathrm{log}}_{4}\left(x\right),$ we will notice $\text{\hspace{0.17em}}a=2.$

This means we will stretch the function $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{4}\left(x\right)\text{\hspace{0.17em}}$ by a factor of 2.

The vertical asymptote is $\text{\hspace{0.17em}}x=0.$

Consider the three key points from the parent function, $\text{\hspace{0.17em}}\left(\frac{1}{4},-1\right),$ $\left(1,0\right),\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(4,1\right).$

The new coordinates are found by multiplying the $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ coordinates by 2.

Label the points $\text{\hspace{0.17em}}\left(\frac{1}{4},-2\right),$ $\left(1,0\right)\text{\hspace{0.17em}},$ and $\text{\hspace{0.17em}}\left(4,\text{2}\right).$

The domain is $\text{\hspace{0.17em}}\left(0,\text{\hspace{0.17em}}\infty \right),$ the range is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),\text{\hspace{0.17em}}$ and the vertical asymptote is $\text{\hspace{0.17em}}x=0.\text{\hspace{0.17em}}$ See [link] .

The domain is $\text{\hspace{0.17em}}\left(0,\infty \right),$ the range is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ and the vertical asymptote is $\text{\hspace{0.17em}}x=0.$

Sketch a graph of $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{2}\text{\hspace{0.17em}}{\mathrm{log}}_{4}\left(x\right)\text{\hspace{0.17em}}$ alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

The domain is $\text{\hspace{0.17em}}\left(0,\infty \right),$ the range is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ and the vertical asymptote is $\text{\hspace{0.17em}}x=0.$

## Combining a shift and a stretch

Sketch a graph of $\text{\hspace{0.17em}}f\left(x\right)=5\mathrm{log}\left(x+2\right).\text{\hspace{0.17em}}$ State the domain, range, and asymptote.

Remember: what happens inside parentheses happens first. First, we move the graph left 2 units, then stretch the function vertically by a factor of 5, as in [link] . The vertical asymptote will be shifted to $\text{\hspace{0.17em}}x=-2.\text{\hspace{0.17em}}$ The x -intercept will be $\text{\hspace{0.17em}}\left(-1,0\right).\text{\hspace{0.17em}}$ The domain will be $\text{\hspace{0.17em}}\left(-2,\infty \right).\text{\hspace{0.17em}}$ Two points will help give the shape of the graph: $\text{\hspace{0.17em}}\left(-1,0\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(8,5\right).\text{\hspace{0.17em}}$ We chose $\text{\hspace{0.17em}}x=8\text{\hspace{0.17em}}$ as the x -coordinate of one point to graph because when $\text{\hspace{0.17em}}x=8,\text{\hspace{0.17em}}$ $\text{\hspace{0.17em}}x+2=10,\text{\hspace{0.17em}}$ the base of the common logarithm.

The domain is $\text{\hspace{0.17em}}\left(-2,\infty \right),$ the range is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ and the vertical asymptote is $\text{\hspace{0.17em}}x=-2.$

Sketch a graph of the function $\text{\hspace{0.17em}}f\left(x\right)=3\mathrm{log}\left(x-2\right)+1.\text{\hspace{0.17em}}$ State the domain, range, and asymptote.

The domain is $\text{\hspace{0.17em}}\left(2,\infty \right),$ the range is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ and the vertical asymptote is $\text{\hspace{0.17em}}x=2.$

## Graphing reflections of f ( x ) = log b ( x )

When the parent function $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ 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. To visualize reflections, we restrict $\text{\hspace{0.17em}}b>1,\text{\hspace{0.17em}}$ and observe the general graph of the parent function $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ alongside the reflection about the x -axis, $\text{\hspace{0.17em}}g\left(x\right)={\mathrm{-log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ and the reflection about the y -axis, $\text{\hspace{0.17em}}h\left(x\right)={\mathrm{log}}_{b}\left(-x\right).$

## Reflections of the parent function y = log b ( x )

The function $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{-log}}_{b}\left(x\right)$

• reflects the parent function $\text{\hspace{0.17em}}y={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ about the x -axis.
• has domain, $\text{\hspace{0.17em}}\left(0,\infty \right),$ range, $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ and vertical asymptote, $\text{\hspace{0.17em}}x=0,$ which are unchanged from the parent function.

The function $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(-x\right)$

• reflects the parent function $\text{\hspace{0.17em}}y={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ about the y -axis.
• has domain $\text{\hspace{0.17em}}\left(-\infty ,0\right).$
• has range, $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ and vertical asymptote, $\text{\hspace{0.17em}}x=0,$ which are unchanged from the parent function.

#### Questions & Answers

f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
Ken Reply
proof
AUSTINE
sebd me some questions about anything ill solve for yall
Manifoldee Reply
how to solve x²=2x+8 factorization?
Kristof Reply
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
SO THE ANSWER IS X=-8
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
1KI POWER 1/3 PLEASE SOLUTIONS
Prashant Reply
hii
Amit
how are you
Dorbor
well
Biswajit
can u tell me concepts
Gaurav
Find the possible value of 8.5 using moivre's theorem
Reuben Reply
which of these functions is not uniformly cintinuous on (0, 1)? sinx
Pooja Reply
which of these functions is not uniformly continuous on 0,1
Basant Reply
solve this equation by completing the square 3x-4x-7=0
Jamiz Reply
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame
what is trigonometry
Jean Reply
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
Rubben Reply
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
goodness. I'm sorry. I will let Alex take the wheel.
Nikki
ouky thanks braa
Rubben
I think he drive me safe
Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
Pab Reply
More example of algebra and trigo
Stephen Reply
What is Indices
Yashim Reply
If one side only of a triangle is given is it possible to solve for the unkown two sides?
Felix Reply
cool
Rubben
kya
Khushnama
please I need help in maths
Dayo Reply
Okey tell me, what's your problem is?
Navin

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