<< Chapter < Page Chapter >> Page >

Given a logarithmic function, identify the domain.

  1. Set up an inequality showing the argument greater than zero.
  2. Solve for x .
  3. Write the domain in interval notation.

Identifying the domain of a logarithmic shift

What is the domain of f ( x ) = log 2 ( x + 3 ) ?

The logarithmic function is defined only when the input is positive, so this function is defined when x + 3 > 0. Solving this inequality,

x + 3 > 0 The input must be positive . x > 3 Subtract 3 .

The domain of f ( x ) = log 2 ( x + 3 ) is ( 3 , ) .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

What is the domain of f ( x ) = log 5 ( x 2 ) + 1 ?

( 2 , )

Got questions? Get instant answers now!

Identifying the domain of a logarithmic shift and reflection

What is the domain of f ( x ) = log ( 5 2 x ) ?

The logarithmic function is defined only when the input is positive, so this function is defined when 5 2 x > 0 . Solving this inequality,

5 2 x > 0 The input must be positive . 2 x > 5 Subtract  5. x < 5 2 Divide by  2  and switch the inequality .

The domain of f ( x ) = log ( 5 2 x ) is ( , 5 2 ) .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

What is the domain of f ( x ) = log ( x 5 ) + 2 ?

( 5 , )

Got questions? Get instant answers now!

Graphing logarithmic functions

Now that we have a feel for the set of values for which a logarithmic function is defined, we move on to graphing logarithmic functions. The family of logarithmic functions includes the parent function y = log b ( x ) along with all its transformations: shifts, stretches, compressions, and reflections.

We begin with the parent function y = log b ( x ) . Because every logarithmic function of this form is the inverse of an exponential function with the form y = b x , their graphs will be reflections of each other across the line y = x . To illustrate this, we can observe the relationship between the input and output values of y = 2 x and its equivalent x = log 2 ( y ) in [link] .

x 3 2 1 0 1 2 3
2 x = y 1 8 1 4 1 2 1 2 4 8
log 2 ( y ) = x 3 2 1 0 1 2 3

Using the inputs and outputs from [link] , we can build another table to observe the relationship between points on the graphs of the inverse functions f ( x ) = 2 x and g ( x ) = log 2 ( x ) . See [link] .

f ( x ) = 2 x ( 3 , 1 8 ) ( 2 , 1 4 ) ( 1 , 1 2 ) ( 0 , 1 ) ( 1 , 2 ) ( 2 , 4 ) ( 3 , 8 )
g ( x ) = log 2 ( x ) ( 1 8 , 3 ) ( 1 4 , 2 ) ( 1 2 , 1 ) ( 1 , 0 ) ( 2 , 1 ) ( 4 , 2 ) ( 8 , 3 )

As we’d expect, the x - and y -coordinates are reversed for the inverse functions. [link] shows the graph of f and g .

Graph of two functions, f(x)=2^x and g(x)=log_2(x), with the line y=x denoting the axis of symmetry.
Notice that the graphs of f ( x ) = 2 x and g ( x ) = log 2 ( x ) are reflections about the line y = x .

Observe the following from the graph:

  • f ( x ) = 2 x has a y -intercept at ( 0 , 1 ) and g ( x ) = log 2 ( x ) has an x - intercept at ( 1 , 0 ) .
  • The domain of f ( x ) = 2 x , ( , ) , is the same as the range of g ( x ) = log 2 ( x ) .
  • The range of f ( x ) = 2 x , ( 0 , ) , is the same as the domain of g ( x ) = log 2 ( x ) .

Characteristics of the graph of the parent function, f ( x ) = log b ( x )

For any real number x and constant b > 0 , b 1 , we can see the following characteristics in the graph of f ( x ) = log b ( x ) :

  • one-to-one function
  • vertical asymptote: x = 0
  • domain: ( 0 , )
  • range: ( , )
  • x- intercept: ( 1 , 0 ) and key point ( b , 1 )
  • y -intercept: none
  • increasing if b > 1
  • decreasing if 0 < b < 1

See [link] .

Two graphs of the function f(x)=log_b(x) with points (1,0) and (b, 1). The first graph shows the line when b>1, and the second graph shows the line when 0<b<1.

[link] shows how changing the base b in f ( x ) = log b ( x ) can affect the graphs. Observe that the graphs compress vertically as the value of the base increases. ( Note: recall that the function ln ( x ) has base e 2 . 718.)

Graph of three equations: y=log_2(x) in blue, y=ln(x) in orange, and y=log(x) in red. The y-axis is the asymptote.
The graphs of three logarithmic functions with different bases, all greater than 1.

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
ok
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

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

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

Ask