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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.

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

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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.

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

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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.

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Algebraic

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

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

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h ( x ) = ln ( 1 2 x )

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

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

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

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

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

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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

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g ( x ) = ln ( 3 x )

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

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

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

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

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

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

f ( x ) = ln ( 2 x )

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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 )

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

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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 )

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

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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

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

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g ( x ) = ln ( x ) 2

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

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

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h ( x ) = 3 ln ( x ) 9

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

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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

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h ( x ) = log 3 4 ( x )

C

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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

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

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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.
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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 )

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

C

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

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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).
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f ( x ) = 2 log ( x )

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f ( x ) = ln ( x )

Graph of f(x)=ln(-x).
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g ( x ) = log ( 4 x + 16 ) + 4

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

Graph of g(x)=log(6-3x)+1.
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h ( x ) = 1 2 ln ( x + 1 ) 3

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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 ) )

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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.
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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 )

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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.
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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

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

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

x 2 .303

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

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

x 0.472

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Extensions

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

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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 ) .

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Prove the conjecture made in the previous exercise.

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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 , ) .

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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.

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Questions & Answers

if tan alpha + beta is equal to sin x + Y then prove that X square + Y square - 2 I got hyperbole 2 Beta + 1 is equal to zero
Rahul Reply
sin^4+sin^2=1, prove that tan^2-tan^4+1=0
SAYANTANI Reply
what is the formula used for this question? "Jamal wants to save $54,000 for a down payment on a home. How much will he need to invest in an account with 8.2% APR, compounding daily, in order to reach his goal in 5 years?"
Kuz Reply
i don't need help solving it I just need a memory jogger please.
Kuz
A = P(1 + r/n) ^rt
Dale
how to solve an expression when equal to zero
Mintah Reply
its a very simple
Kavita
gave your expression then i solve
Kavita
Hy guys, I have a problem when it comes on solving equations and expressions, can you help me 😭😭
Thuli
Tomorrow its an revision on factorising and Simplifying...
Thuli
ok sent the quiz
kurash
send
Kavita
Hi
Masum
What is the value of log-1
Masum
the value of log1=0
Kavita
Log(-1)
Masum
What is the value of i^i
Masum
log -1 is 1.36
kurash
No
Masum
no I m right
Kavita
No sister.
Masum
no I m right
Kavita
tan20°×tan30°×tan45°×tan50°×tan60°×tan70°
Joju Reply
jaldi batao
Joju
Find the value of x between 0degree and 360 degree which satisfy the equation 3sinx =tanx
musah Reply
what is sine?
tae Reply
what is the standard form of 1
Sanjana Reply
1×10^0
Akugry
Evalute exponential functions
Sujata Reply
30
Shani
The sides of a triangle are three consecutive natural number numbers and it's largest angle is twice the smallest one. determine the sides of a triangle
Jaya Reply
Will be with you shortly
Inkoom
3, 4, 5 principle from geo? sounds like a 90 and 2 45's to me that my answer
Neese
answer is 2, 3, 4
Gaurav
prove that [a+b, b+c, c+a]= 2[a b c]
Ashutosh Reply
can't prove
Akugry
i can prove [a+b+b+c+c+a]=2[a+b+c]
this is simple
Akugry
hi
Stormzy
x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
HERVE Reply
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
Oliver Reply
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
Kwesi Reply
don't you know?
Inkoom
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
Martina Reply

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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