<< 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 do I set up the problem?
Harshika Reply
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
hello, I am happy to help!
Shirley Reply
please can go further on polynomials quadratic
Abdullahi
hi mam
Mark
I need quadratic equation link to Alpa Beta
Abdullahi Reply
find the value of 2x=32
Felix Reply
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
yes i wantt to review
Mark
use the y -intercept and slope to sketch the graph of the equation y=6x
Only Reply
how do we prove the quadratic formular
Seidu Reply
please help me prove quadratic formula
Darius
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
Shirley Reply
thank you help me with how to prove the quadratic equation
Seidu
may God blessed u for that. Please I want u to help me in sets.
Opoku
what is math number
Tric Reply
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
can you teacch how to solve that🙏
Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply

Get the best College algebra course in your pocket!





Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

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

Ask