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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

The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit 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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