# 4.4 Graphs of logarithmic functions  (Page 3/8)

 Page 3 / 8

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

1. Draw and label the vertical asymptote, $\text{\hspace{0.17em}}x=0.$
2. Plot the x- intercept, $\text{\hspace{0.17em}}\left(1,0\right).$
3. Plot the key point $\text{\hspace{0.17em}}\left(b,1\right).$
4. Draw a smooth curve through the points.
5. State the domain, $\text{\hspace{0.17em}}\left(0,\infty \right),$ the range, $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ and the vertical asymptote, $\text{\hspace{0.17em}}x=0.$

## Graphing a logarithmic function with the form f ( x ) = log b ( x ).

Graph $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{5}\left(x\right).\text{\hspace{0.17em}}$ State the domain, range, and asymptote.

Before graphing, identify the behavior and key points for the graph.

• Since $\text{\hspace{0.17em}}b=5\text{\hspace{0.17em}}$ is greater than one, we know the function is increasing. The left tail of the graph will approach the vertical asymptote $\text{\hspace{0.17em}}x=0,$ and the right tail will increase slowly without bound.
• The x -intercept is $\text{\hspace{0.17em}}\left(1,0\right).$
• The key point $\text{\hspace{0.17em}}\left(5,1\right)\text{\hspace{0.17em}}$ is on the graph.
• We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points (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 $\text{\hspace{0.17em}}x=0.$

Graph $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{\frac{1}{5}}\left(x\right).\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}x=0.$

## Graphing transformations of logarithmic functions

As we mentioned in the beginning of the section, transformations of logarithmic graphs behave similarly to those of other parent functions. We can shift, stretch, compress, and reflect the parent function $\text{\hspace{0.17em}}y={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ without loss of shape.

## Graphing a horizontal shift of f ( x ) = log b ( x )

When a constant $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ is added to the input of the parent function $\text{\hspace{0.17em}}f\left(x\right)=lo{g}_{b}\left(x\right),$ the result is a horizontal shift     $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ units in the opposite direction of the sign on $\text{\hspace{0.17em}}c.\text{\hspace{0.17em}}$ To visualize horizontal shifts, we can observe the general graph of the parent function $f\left(x\right)={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ and for $\text{\hspace{0.17em}}c>0\text{\hspace{0.17em}}$ alongside the shift left, $\text{\hspace{0.17em}}g\left(x\right)={\mathrm{log}}_{b}\left(x+c\right),$ and the shift right, $\text{\hspace{0.17em}}h\left(x\right)={\mathrm{log}}_{b}\left(x-c\right).$ See [link] .

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

For any constant $\text{\hspace{0.17em}}c,$ the function $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x+c\right)$

• shifts the parent function $\text{\hspace{0.17em}}y={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ left $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ units if $\text{\hspace{0.17em}}c>0.$
• shifts the parent function $\text{\hspace{0.17em}}y={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ right $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ units if $\text{\hspace{0.17em}}c<0.$
• has the vertical asymptote $\text{\hspace{0.17em}}x=-c.$
• has domain $\text{\hspace{0.17em}}\left(-c,\infty \right).$
• has range $\text{\hspace{0.17em}}\left(-\infty ,\infty \right).$

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

1. Identify the horizontal shift:
1. If $\text{\hspace{0.17em}}c>0,$ shift the graph of $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ left $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ units.
2. If $\text{\hspace{0.17em}}c<0,$ shift the graph of $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ right $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ units.
2. Draw the vertical asymptote $\text{\hspace{0.17em}}x=-c.$
3. Identify three key points from the parent function. Find new coordinates for the shifted functions by subtracting $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ from the $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ coordinate.
4. Label the three points.
5. The Domain is $\text{\hspace{0.17em}}\left(-c,\infty \right),$ the range is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ and the vertical asymptote is $\text{\hspace{0.17em}}x=-c.$

## Graphing a horizontal shift of the parent function y = log b ( x )

Sketch the horizontal shift $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{3}\left(x-2\right)\text{\hspace{0.17em}}$ alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.

Since the function is $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{3}\left(x-2\right),$ we notice $\text{\hspace{0.17em}}x+\left(-2\right)=x–2.$

Thus $\text{\hspace{0.17em}}c=-2,$ so $\text{\hspace{0.17em}}c<0.\text{\hspace{0.17em}}$ This means we will shift the function $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{3}\left(x\right)\text{\hspace{0.17em}}$ right 2 units.

The vertical asymptote is $\text{\hspace{0.17em}}x=-\left(-2\right)\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}x=2.$

Consider the three key points from the parent function, $\text{\hspace{0.17em}}\left(\frac{1}{3},-1\right),$ $\left(1,0\right),$ and $\text{\hspace{0.17em}}\left(3,1\right).$

The new coordinates are found by adding 2 to the $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ coordinates.

Label the points $\text{\hspace{0.17em}}\left(\frac{7}{3},-1\right),$ $\left(3,0\right),$ and $\text{\hspace{0.17em}}\left(5,1\right).$

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 $\text{\hspace{0.17em}}x=2.$

how can are find the domain and range of a relations
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
can I see the picture
How would you find if a radical function is one to one?
how to understand calculus?
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
what is foci?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
i want to sure my answer of the exercise
what is the diameter of(x-2)²+(y-3)²=25
how to solve the Identity ?
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
what is a complex number used for?
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim