<< 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

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
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