<< 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

how to solve the Identity ?
Barcenas Reply
what type of identity
Jeffrey
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.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
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
Is there any rule we can use to get the nth term ?
Anwar Reply
how do you get the (1.4427)^t in the carp problem?
Gabrielle Reply
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Sandra Reply
Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
prince Reply
hello
Jessica Reply
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of ?105°F??105°F? occurs at 5PM and the average temperature for the day is ?85°F.??85°F.? Find the temperature, to the nearest degree, at 9AM.
Karlee Reply
if you have the amplitude and the period and the phase shift ho would you know where to start and where to end?
Jean Reply
rotation by 80 of (x^2/9)-(y^2/16)=1
Garrett Reply
thanks the domain is good but a i would like to get some other examples of how to find the range of a function
bashiir Reply
what is the standard form if the focus is at (0,2) ?
Lorejean Reply

Get the best Precalculus course in your pocket!





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