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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

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
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Shirleen Reply
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Adu
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
kinnecy Reply
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
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Abdirahman Reply

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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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