<< Chapter < Page Chapter >> Page >
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.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

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

x = 1

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

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
prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x
Rockstar Reply
tanh`(x-iy) =A+iB, find A and B
Pankaj Reply
B=Ai-itan(hx-hiy)
Rukmini
what is the addition of 101011 with 101010
Branded Reply
If those numbers are binary, it's 1010101. If they are base 10, it's 202021.
Jack
extra power 4 minus 5 x cube + 7 x square minus 5 x + 1 equal to zero
archana Reply
the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
Lhorren Reply
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
Vinay Reply
10y
Michael

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask