<< Chapter < Page Chapter >> Page >

Section exercises

Verbal

The inverse of every logarithmic function is an exponential function and vice-versa. What does this tell us about the relationship between the coordinates of the points on the graphs of each?

Since the functions are inverses, their graphs are mirror images about the line y = x . So for every point ( a , b ) on the graph of a logarithmic function, there is a corresponding point ( b , a ) on the graph of its inverse exponential function.

Got questions? Get instant answers now!

What type(s) of translation(s), if any, affect the range of a logarithmic function?

Got questions? Get instant answers now!

What type(s) of translation(s), if any, affect the domain of a logarithmic function?

Shifting the function right or left and reflecting the function about the y-axis will affect its domain.

Got questions? Get instant answers now!

Consider the general logarithmic function f ( x ) = log b ( x ) . Why can’t x be zero?

Got questions? Get instant answers now!

Does the graph of a general logarithmic function have a horizontal asymptote? Explain.

No. A horizontal asymptote would suggest a limit on the range, and the range of any logarithmic function in general form is all real numbers.

Got questions? Get instant answers now!

Algebraic

For the following exercises, state the domain and range of the function.

f ( x ) = log 3 ( x + 4 )

Got questions? Get instant answers now!

h ( x ) = ln ( 1 2 x )

Domain: ( , 1 2 ) ; Range: ( , )

Got questions? Get instant answers now!

g ( x ) = log 5 ( 2 x + 9 ) 2

Got questions? Get instant answers now!

h ( x ) = ln ( 4 x + 17 ) 5

Domain: ( 17 4 , ) ; Range: ( , )

Got questions? Get instant answers now!

f ( x ) = log 2 ( 12 3 x ) 3

Got questions? Get instant answers now!

For the following exercises, state the domain and the vertical asymptote of the function.

f ( x ) = log b ( x 5 )

Domain: ( 5 , ) ; Vertical asymptote: x = 5

Got questions? Get instant answers now!

g ( x ) = ln ( 3 x )

Got questions? Get instant answers now!

f ( x ) = log ( 3 x + 1 )

Domain: ( 1 3 , ) ; Vertical asymptote: x = 1 3

Got questions? Get instant answers now!

f ( x ) = 3 log ( x ) + 2

Got questions? Get instant answers now!

g ( x ) = ln ( 3 x + 9 ) 7

Domain: ( 3 , ) ; Vertical asymptote: x = 3

Got questions? Get instant answers now!

For the following exercises, state the domain, vertical asymptote, and end behavior of the function.

f ( x ) = ln ( 2 x )

Got questions? Get instant answers now!

f ( x ) = log ( x 3 7 )

Domain: ( 3 7 , ) ;
Vertical asymptote: x = 3 7 ; End behavior: as x ( 3 7 ) + , f ( x ) and as x , f ( x )

Got questions? Get instant answers now!

h ( x ) = log ( 3 x 4 ) + 3

Got questions? Get instant answers now!

g ( x ) = ln ( 2 x + 6 ) 5

Domain: ( 3 , ) ; Vertical asymptote: x = 3 ;
End behavior: as x 3 + , f ( x ) and as x , f ( x )

Got questions? Get instant answers now!

f ( x ) = log 3 ( 15 5 x ) + 6

Got questions? Get instant answers now!

For the following exercises, state the domain, range, and x - and y -intercepts, if they exist. If they do not exist, write DNE.

h ( x ) = log 4 ( x 1 ) + 1

Domain: ( 1 , ) ; Range: ( , ) ; Vertical asymptote: x = 1 ; x -intercept: ( 5 4 , 0 ) ; y -intercept: DNE

Got questions? Get instant answers now!

f ( x ) = log ( 5 x + 10 ) + 3

Got questions? Get instant answers now!

g ( x ) = ln ( x ) 2

Domain: ( , 0 ) ; Range: ( , ) ; Vertical asymptote: x = 0 ; x -intercept: ( e 2 , 0 ) ; y -intercept: DNE

Got questions? Get instant answers now!

f ( x ) = log 2 ( x + 2 ) 5

Got questions? Get instant answers now!

h ( x ) = 3 ln ( x ) 9

Domain: ( 0 , ) ; Range: ( , ) ; Vertical asymptote: x = 0 ; x -intercept: ( e 3 , 0 ) ; y -intercept: DNE

Got questions? Get instant answers now!

Graphical

For the following exercises, match each function in [link] with the letter corresponding to its graph.

Graph of five logarithmic functions.

For the following exercises, match each function in [link] with the letter corresponding to its graph.

Graph of three logarithmic functions.

f ( x ) = log 1 3 ( x )

B

Got questions? Get instant answers now!

h ( x ) = log 3 4 ( x )

C

Got questions? Get instant answers now!

For the following exercises, sketch the graphs of each pair of functions on the same axis.

f ( x ) = log ( x ) and g ( x ) = 10 x

Got questions? Get instant answers now!

f ( x ) = log ( x ) and g ( x ) = log 1 2 ( x )

Graph of two functions, g(x) = log_(1/2)(x) in orange and f(x)=log(x) in blue.
Got questions? Get instant answers now!

f ( x ) = log 4 ( x ) and g ( x ) = ln ( x )

Got questions? Get instant answers now!

f ( x ) = e x and g ( x ) = ln ( x )

Graph of two functions, g(x) = ln(1/2)(x) in orange and f(x)=e^(x) in blue.
Got questions? Get instant answers now!

For the following exercises, match each function in [link] with the letter corresponding to its graph.

Graph of three logarithmic functions.

f ( x ) = log 4 ( x + 2 )

Got questions? Get instant answers now!

g ( x ) = log 4 ( x + 2 )

C

Got questions? Get instant answers now!

h ( x ) = log 4 ( x + 2 )

Got questions? Get instant answers now!

For the following exercises, sketch the graph of the indicated function.

f ( x ) = log 2 ( x + 2 )

Graph of f(x)=log_2(x+2).
Got questions? Get instant answers now!

f ( x ) = 2 log ( x )

Got questions? Get instant answers now!

f ( x ) = ln ( x )

Graph of f(x)=ln(-x).
Got questions? Get instant answers now!

g ( x ) = log ( 4 x + 16 ) + 4

Got questions? Get instant answers now!

g ( x ) = log ( 6 3 x ) + 1

Graph of g(x)=log(6-3x)+1.
Got questions? Get instant answers now!

h ( x ) = 1 2 ln ( x + 1 ) 3

Got questions? Get instant answers now!

For the following exercises, write a logarithmic equation corresponding to the graph shown.

Use y = log 2 ( x ) as the parent function.

The graph y=log_2(x) has been reflected over the y-axis and shifted to the right by 1.

f ( x ) = log 2 ( ( x 1 ) )

Got questions? Get instant answers now!

Use f ( x ) = log 3 ( x ) as the parent function.

The graph y=log_3(x) has been reflected over the x-axis, vertically stretched by 3, and shifted to the left by 4.
Got questions? Get instant answers now!

Use f ( x ) = log 4 ( x ) as the parent function.

The graph y=log_4(x) has been vertically stretched by 3, and shifted to the left by 2.

f ( x ) = 3 log 4 ( x + 2 )

Got questions? Get instant answers now!

Use f ( x ) = log 5 ( x ) as the parent function.

The graph y=log_3(x) has been reflected over the x-axis and y-axis, vertically stretched by 2, and shifted to the right by 5.
Got questions? Get instant answers now!

Technology

For the following exercises, use a graphing calculator to find approximate solutions to each equation.

log ( x 1 ) + 2 = ln ( x 1 ) + 2

x = 2

Got questions? Get instant answers now!

log ( 2 x 3 ) + 2 = log ( 2 x 3 ) + 5

Got questions? Get instant answers now!

ln ( x 2 ) = ln ( x + 1 )

x 2 .303

Got questions? Get instant answers now!

2 ln ( 5 x + 1 ) = 1 2 ln ( 5 x ) + 1

Got questions? Get instant answers now!

1 3 log ( 1 x ) = log ( x + 1 ) + 1 3

x 0.472

Got questions? Get instant answers now!

Extensions

Let b be any positive real number such that b 1. What must log b 1 be equal to? Verify the result.

Got questions? Get instant answers now!

Explore and discuss the graphs of f ( x ) = log 1 2 ( x ) and g ( x ) = log 2 ( x ) . Make a conjecture based on the result.

The graphs of f ( x ) = log 1 2 ( x ) and g ( x ) = log 2 ( x ) appear to be the same; Conjecture: for any positive base b 1 , log b ( x ) = log 1 b ( x ) .

Got questions? Get instant answers now!

Prove the conjecture made in the previous exercise.

Got questions? Get instant answers now!

What is the domain of the function f ( x ) = ln ( x + 2 x 4 ) ? Discuss the result.

Recall that the argument of a logarithmic function must be positive, so we determine where x + 2 x 4 > 0 . From the graph of the function f ( x ) = x + 2 x 4 , note that the graph lies above the x -axis on the interval ( , 2 ) and again to the right of the vertical asymptote, that is ( 4 , ) . Therefore, the domain is ( , 2 ) ( 4 , ) .

Got questions? Get instant answers now!

Use properties of exponents to find the x -intercepts of the function f ( x ) = log ( x 2 + 4 x + 4 ) algebraically. Show the steps for solving, and then verify the result by graphing the function.

Got questions? Get instant answers now!

Questions & Answers

what is mutation
Janga Reply
what is a cell
Sifune Reply
how is urine form
Sifune
what is antagonism?
mahase Reply
classification of plants, gymnosperm features.
Linsy Reply
what is the features of gymnosperm
Linsy
how many types of solid did we have
Samuel Reply
what is an ionic bond
Samuel
What is Atoms
Daprince Reply
what is fallopian tube
Merolyn
what is bladder
Merolyn
what's bulbourethral gland
Eduek Reply
urine is formed in the nephron of the renal medulla in the kidney. It starts from filtration, then selective reabsorption and finally secretion
onuoha Reply
State the evolution relation and relevance between endoplasmic reticulum and cytoskeleton as it relates to cell.
Jeremiah
what is heart
Konadu Reply
how is urine formed in human
Konadu
how is urine formed in human
Rahma
what is the diference between a cavity and a canal
Pelagie Reply
what is the causative agent of malaria
Diamond
malaria is caused by an insect called mosquito.
Naomi
Malaria is cause by female anopheles mosquito
Isaac
Malaria is caused by plasmodium Female anopheles mosquitoe is d carrier
Olalekan
a canal is more needed in a root but a cavity is a bad effect
Commander
what are pathogens
Don Reply
In biology, a pathogen (Greek: πάθος pathos "suffering", "passion" and -γενής -genēs "producer of") in the oldest and broadest sense, is anything that can produce disease. A pathogen may also be referred to as an infectious agent, or simply a germ. The term pathogen came into use in the 1880s.[1][2
Zainab
A virus
Commander
Definition of respiration
Muhsin Reply
respiration is the process in which we breath in oxygen and breath out carbon dioxide
Achor
how are lungs work
Commander
where does digestion begins
Achiri Reply
in the mouth
EZEKIEL
what are the functions of follicle stimulating harmones?
Rashima Reply
stimulates the follicle to release the mature ovum into the oviduct
Davonte
what are the functions of Endocrine and pituitary gland
Chinaza
endocrine secrete hormone and regulate body process
Achor
while pituitary gland is an example of endocrine system and it's found in the Brain
Achor
what's biology?
Egbodo Reply
Biology is the study of living organisms, divided into many specialized field that cover their morphology, physiology,anatomy, behaviour,origin and distribution.
Lisah
biology is the study of life.
Alfreda
Biology is the study of how living organisms live and survive in a specific environment
Sifune
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, 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