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Translations of the Parent Function y = log b ( x )
Translation Form
Shift
  • Horizontally c units to the left
  • Vertically d units up
y = log b ( x + c ) + d
Stretch and Compress
  • Stretch if | a | > 1
  • Compression if | a | < 1
y = a log b ( x )
Reflect about the x -axis y = log b ( x )
Reflect about the y -axis y = log b ( x )
General equation for all translations y = a log b ( x + c ) + d

Translations of logarithmic functions

All translations of the parent logarithmic function, y = log b ( x ) , have the form

  f ( x ) = a log b ( x + c ) + d

where the parent function, y = log b ( x ) , b > 1 , is

  • shifted vertically up d units.
  • shifted horizontally to the left c units.
  • stretched vertically by a factor of | a | if | a | > 0.
  • compressed vertically by a factor of | a | if 0 < | a | < 1.
  • reflected about the x- axis when a < 0.

For f ( x ) = log ( x ) , the graph of the parent function is reflected about the y -axis.

Finding the vertical asymptote of a logarithm graph

What is the vertical asymptote of f ( x ) = −2 log 3 ( x + 4 ) + 5 ?

The vertical asymptote is at x = 4.

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What is the vertical asymptote of f ( x ) = 3 + ln ( x 1 ) ?

x = 1

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Finding the equation from a graph

Find a possible equation for the common logarithmic function graphed in [link] .

Graph of a logarithmic function with a vertical asymptote at x=-2, has been vertically reflected, and passes through the points (-1, 1) and (2, -1).

This graph has a vertical asymptote at x = –2 and has been vertically reflected. We do not know yet the vertical shift or the vertical stretch. We know so far that the equation will have form:

f ( x ) = a log ( x + 2 ) + k

It appears the graph passes through the points ( –1 , 1 ) and ( 2 , –1 ) . Substituting ( –1 , 1 ) ,

1 = a log ( −1 + 2 ) + k Substitute  ( −1 , 1 ) . 1 = a log ( 1 ) + k Arithmetic . 1 = k log(1) = 0.

Next, substituting in ( 2 , –1 ) ,

1 = a log ( 2 + 2 ) + 1 Plug in  ( 2 , −1 ) . 2 = a log ( 4 ) Arithmetic .    a = 2 log ( 4 ) Solve for  a .

This gives us the equation f ( x ) = 2 log ( 4 ) log ( x + 2 ) + 1.

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Give the equation of the natural logarithm graphed in [link] .

Graph of a logarithmic function with a vertical asymptote at x=-3, has been vertically stretched by 2, and passes through the points (-1, -1).

f ( x ) = 2 ln ( x + 3 ) 1

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Is it possible to tell the domain and range and describe the end behavior of a function just by looking at the graph?

Yes, if we know the function is a general logarithmic function. For example, look at the graph in [link] . The graph approaches x = −3 (or thereabouts) more and more closely, so x = −3 is, or is very close to, the vertical asymptote. It approaches from the right, so the domain is all points to the right, { x | x > −3 } . The range, as with all general logarithmic functions, is all real numbers. And we can see the end behavior because the graph goes down as it goes left and up as it goes right. The end behavior is that as x 3 + , f ( x ) and as x , f ( x ) .

Access these online resources for additional instruction and practice with graphing logarithms.

Key equations

General Form for the Translation of the Parent Logarithmic Function   f ( x ) = log b ( x )   f ( x ) = a log b ( x + c ) + d

Key concepts

  • To find the domain of a logarithmic function, set up an inequality showing the argument greater than zero, and solve for x . See [link] and [link]
  • The graph of the parent function f ( x ) = log b ( x ) has an x- intercept at ( 1 , 0 ) , domain ( 0 , ) , range ( , ) , vertical asymptote x = 0 , and
    • if b > 1 , the function is increasing.
    • if 0 < b < 1 , the function is decreasing.
    See [link] .
  • The equation f ( x ) = log b ( x + c ) shifts the parent function y = log b ( x ) horizontally
    • left c units if c > 0.
    • right c units if c < 0.
    See [link] .
  • The equation f ( x ) = log b ( x ) + d shifts the parent function y = log b ( x ) vertically
    • up d units if d > 0.
    • down d units if d < 0.
    See [link] .
  • For any constant a > 0 , the equation f ( x ) = a log b ( x )
    • stretches the parent function y = log b ( x ) vertically by a factor of a if | a | > 1.
    • compresses the parent function y = log b ( x ) vertically by a factor of a if | a | < 1.
    See [link] and [link] .
  • When the parent function y = log b ( x ) is multiplied by 1 , the result is a reflection about the x -axis. When the input is multiplied by 1 , the result is a reflection about the y -axis.
    • The equation f ( x ) = log b ( x ) represents a reflection of the parent function about the x- axis.
    • The equation f ( x ) = log b ( x ) represents a reflection of the parent function about the y- axis.
    See [link] .
    • A graphing calculator may be used to approximate solutions to some logarithmic equations See [link] .
  • All translations of the logarithmic function can be summarized by the general equation   f ( x ) = a log b ( x + c ) + d . See [link] .
  • Given an equation with the general form   f ( x ) = a log b ( x + c ) + d , we can identify the vertical asymptote x = c for the transformation. See [link] .
  • Using the general equation f ( x ) = a log b ( x + c ) + d , we can write the equation of a logarithmic function given its graph. See [link] .

Questions & Answers

Cos45/sec30+cosec30=
dinesh Reply
Cos 45 = 1/ √ 2 sec 30 = 2/√3 cosec 30 = 2. =1/√2 / 2/√3+2 =1/√2/2+2√3/√3 =1/√2*√3/2+2√3 =√3/√2(2+2√3) =√3/2√2+2√6 --------- (1) =√3 (2√6-2√2)/((2√6)+2√2))(2√6-2√2) =2√3(√6-√2)/(2√6)²-(2√2)² =2√3(√6-√2)/24-8 =2√3(√6-√2)/16 =√18-√16/8 =3√2-√6/8 ----------(2)
exercise 1.2 solution b....isnt it lacking
Miiro Reply
I dnt get dis work well
john Reply
what is one-to-one function
Iwori Reply
what is the procedure in solving quadratic equetion at least 6?
Qhadz Reply
Almighty formula or by factorization...or by graphical analysis
Damian
I need to learn this trigonometry from A level.. can anyone help here?
wisdom Reply
yes am hia
Miiro
tanh2x =2tanhx/1+tanh^2x
Gautam Reply
cos(a+b)+cos(a-b)/sin(a+b)-sin(a-b)=cotb ... pls some one should help me with this..thanks in anticipation
favour Reply
f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
Ken Reply
proof
AUSTINE
sebd me some questions about anything ill solve for yall
Manifoldee Reply
cos(a+b)+cos(a-b)/sin(a+b)-sin(a-b)= cotb
favour
how to solve x²=2x+8 factorization?
Kristof Reply
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
SO THE ANSWER IS X=-8
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
i am in
Cliff
1KI POWER 1/3 PLEASE SOLUTIONS
Prashant Reply
hii
Amit
how are you
Dorbor
well
Biswajit
can u tell me concepts
Gaurav
Find the possible value of 8.5 using moivre's theorem
Reuben Reply
which of these functions is not uniformly cintinuous on (0, 1)? sinx
Pooja Reply
helo
Akash
hlo
Akash
Hello
Hudheifa
which of these functions is not uniformly continuous on 0,1
Basant 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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