4.4 Graphs of logarithmic functions (Page 3/8)

<< Chapter < Page

Precalculus

Page 3 / 8

Chapter >> Page >

How To

Given a logarithmic function with the form $f (x) = \log_{b} (x),$ graph the function.

Draw and label the vertical asymptote, $x = 0.$
Plot the x- intercept, $(1, 0) .$
Plot the key point $(b, 1) .$
Draw a smooth curve through the points.
State the domain, $(0, \infty),$ the range, $(- \infty, \infty),$ and the vertical asymptote, $x = 0.$

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

Graph $f (x) = \log_{5} (x) .$ State the domain, range, and asymptote.

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

Since $b = 5$ is greater than one, we know the function is increasing. The left tail of the graph will approach the vertical asymptote $x = 0,$ and the right tail will increase slowly without bound.
The x -intercept is $(1, 0) .$
The key point $(5, 1)$ 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] ).

Graph of f(x)=log_5(x) with labeled points at (1, 0) and (5, 1). The y-axis is the asymptote.

The domain is $(0, \infty),$ the range is $(- \infty, \infty),$ and the vertical asymptote is $x = 0.$

Got questions? Get instant answers now!

Try It

Graph $f (x) = \log_{\frac{1}{5}} (x) .$ State the domain, range, and asymptote.

Graph of f(x)=log_(1/5)(x) with labeled points at (1/5, 1) and (1, 0). The y-axis is the asymptote.

The domain is $(0, \infty),$ the range is $(- \infty, \infty),$ and the vertical asymptote is $x = 0.$

Got questions? Get instant answers now!

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 $y = \log_{b} (x)$ without loss of shape.

Graphing a horizontal shift of f ( x ) = log _b ( x )

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

Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0 and g(x)=log_b(x+c) is the translation function with an asymptote at x=-c. This shows the translation of shifting left.

A General Note

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

For any constant $c,$ the function $f (x) = \log_{b} (x + c)$

shifts the parent function $y = \log_{b} (x)$ left $c$ units if $c > 0.$
shifts the parent function $y = \log_{b} (x)$ right $c$ units if $c < 0.$
has the vertical asymptote $x = - c .$
has domain $(- c, \infty) .$
has range $(- \infty, \infty) .$

How To

Given a logarithmic function with the form $f (x) = \log_{b} (x + c),$ graph the translation.

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

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

Sketch the horizontal shift $f (x) = \log_{3} (x - 2)$ alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.

Since the function is $f (x) = \log_{3} (x - 2),$ we notice $x + (- 2) = x - 2.$

Thus $c = - 2,$ so $c < 0.$ This means we will shift the function $f (x) = \log_{3} (x)$ right 2 units.

The vertical asymptote is $x = - (- 2)$ or $x = 2.$

Consider the three key points from the parent function, $(\frac{1}{3}, −1),$ $(1, 0),$ and $(3, 1) .$

The new coordinates are found by adding 2 to the $x$ coordinates.

Label the points $(\frac{7}{3}, −1),$ $(3, 0),$ and $(5, 1) .$

The domain is $(2, \infty),$ the range is $(- \infty, \infty),$ and the vertical asymptote is $x = 2.$

Graph of two functions. The parent function is y=log_3(x), with an asymptote at x=0 and labeled points at (1/3, -1), (1, 0), and (3, 1).The translation function f(x)=log_3(x-2) has an asymptote at x=2 and labeled points at (3, 0) and (5, 1).

Got questions? Get instant answers now!

Questions & Answers

how does Neisseria cause meningitis

Nyibol Reply

what is microbiologist

Muhammad Reply

what is errata

Muhammad

is the branch of biology that deals with the study of microorganisms.

Ntefuni Reply

What is microbiology

Mercy Reply

studies of microbes

Louisiaste

when we takee the specimen which lumbar,spin,

Ziyad Reply

How bacteria create energy to survive?

Muhamad Reply

Bacteria doesn't produce energy they are dependent upon their substrate in case of lack of nutrients they are able to make spores which helps them to sustain in harsh environments

_Adnan

But not all bacteria make spores, l mean Eukaryotic cells have Mitochondria which acts as powerhouse for them, since bacteria don't have it, what is the substitution for it?

Muhamad

they make spores

Louisiaste

what is sporadic nd endemic, epidemic

Aminu Reply

the significance of food webs for disease transmission

Abreham

food webs brings about an infection as an individual depends on number of diseased foods or carriers dully.

Mark

explain assimilatory nitrate reduction

Esinniobiwa Reply

Assimilatory nitrate reduction is a process that occurs in some microorganisms, such as bacteria and archaea, in which nitrate (NO3-) is reduced to nitrite (NO2-), and then further reduced to ammonia (NH3).

Elkana

This process is called assimilatory nitrate reduction because the nitrogen that is produced is incorporated in the cells of microorganisms where it can be used in the synthesis of amino acids and other nitrogen products

Elkana

Examples of thermophilic organisms

Shu Reply

Give Examples of thermophilic organisms

Shu

advantages of normal Flora to the host

Micheal Reply

Prevent foreign microbes to the host

Abubakar

they provide healthier benefits to their hosts

ayesha

They are friends to host only when Host immune system is strong and become enemies when the host immune system is weakened . very bad relationship!

Mark

what is cell

faisal Reply

cell is the smallest unit of life

Fauziya

cell is the smallest unit of life

Akanni

Innocent

cell is the structural and functional unit of life

Hasan

is the fundamental units of Life

Musa

what are emergency diseases

Micheal Reply

There are nothing like emergency disease but there are some common medical emergency which can occur simultaneously like Bleeding,heart attack,Breathing difficulties,severe pain heart stock.Hope you will get my point .Have a nice day ❣️

_Adnan

define infection ,prevention and control

Innocent

I think infection prevention and control is the avoidance of all things we do that gives out break of infections and promotion of health practices that promote life

Lubega

Heyy Lubega hussein where are u from?

_Adnan

en français

Adama

which site have a normal flora

ESTHER Reply

Many sites of the body have it Skin Nasal cavity Oral cavity Gastro intestinal tract

Safaa

skin

Asiina

skin,Oral,Nasal,GIt

Sadik

How can Commensal can Bacteria change into pathogen?

Sadik

How can Commensal Bacteria change into pathogen?

Sadik

all

Tesfaye

by fussion

Asiina

what are the advantages of normal Flora to the host

Micheal

what are the ways of control and prevention of nosocomial infection in the hospital

Micheal

what is inflammation

Shelly Reply

part of a tissue or an organ being wounded or bruised.

Wilfred

what term is used to name and classify microorganisms?

Micheal Reply

Binomial nomenclature

adeolu

Got questions? Join the online conversation and get instant answers!

Jobilize.com Reply

<< Chapter < Page Page > Chapter >>

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

Source: OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Precalculus' conversation and receive update notifications?

Ask

	4 Graphing a horizontal shift of f ( x ) = log b ( x ) By OpenStax Read Online Course
	2 Graphing logarithmic functions By OpenStax Read Online Course
©flickr:	Morphology By Jugnu Khan Start Quiz
	Business Law Ethics By Kevin Moquin Start Quiz
	English Proficiency Test By Anindyo Mukhopadhyay Start Quiz
	English Vocabulary By Jordon Humphreys Start Quiz
	1 Pharmacology Nervous System MCQ By Rohini Ajay Start Quiz
	Nutrition Exam By Hannah Sheth Start Quiz
	Anthropology Language Culture By Richley Crapo Start Assignment
	6 Sociology 06 Groups and Organization MCQ By OpenStax Start Quiz
	Cultural Anthropology Definition By Richley Crapo Start Assignment
	8 BOD- Cardio Quiz By Brooke Delaney Start Exam

4.4 Graphs of logarithmic functions (Page 3/8)

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

Graphing transformations of logarithmic functions

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

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

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

Questions & Answers

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

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

Graphing a horizontal shift of f ( x ) = log _b ( x )

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

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