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Given a logarithmic function with the form f ( x ) = log b ( x ) , graph the function.

  1. Draw and label the vertical asymptote, x = 0.
  2. Plot the x- intercept, ( 1 , 0 ) .
  3. Plot the key point ( b , 1 ) .
  4. Draw a smooth curve through the points.
  5. State the domain, ( 0 , ) , the range, ( , ) , and the vertical asymptote, x = 0.

Graphing a logarithmic function with the form f ( x ) = log b ( x ).

Graph f ( x ) = log 5 ( x ) . State the domain, range, and asymptote.

Before graphing, identify the behavior and key points for the graph.

  • Since b = 5 is greater than one, we know the function is increasing. The left tail of the graph will approach the vertical asymptote x = 0 , and the right tail will increase slowly without bound.
  • The x -intercept is ( 1 , 0 ) .
  • The key point ( 5 , 1 ) is on the graph.
  • We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points (see [link] ).
Graph of f(x)=log_5(x) with labeled points at (1, 0) and (5, 1). The y-axis is the asymptote.

The domain is ( 0 , ) , the range is ( , ) , and the vertical asymptote is x = 0.

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Graph f ( x ) = log 1 5 ( x ) . State the domain, range, and asymptote.

Graph of f(x)=log_(1/5)(x) with labeled points at (1/5, 1) and (1, 0). The y-axis is the asymptote.

The domain is ( 0 , ) , the range is ( , ) , and the vertical asymptote is x = 0.

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Graphing transformations of logarithmic functions

As we mentioned in the beginning of the section, transformations of logarithmic graphs behave similarly to those of other parent functions. We can shift, stretch, compress, and reflect the parent function y = log b ( x ) without loss of shape.

Graphing a horizontal shift of f ( x ) = log b ( x )

When a constant c is added to the input of the parent function f ( x ) = l o g b ( x ) , the result is a horizontal shift     c units in the opposite direction of the sign on c . To visualize horizontal shifts, we can observe the general graph of the parent function f ( x ) = log b ( x ) and for c > 0 alongside the shift left, g ( x ) = log b ( x + c ) , and the shift right, h ( x ) = log b ( x c ) . See [link] .

Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0  and g(x)=log_b(x+c) is the translation function with an asymptote at x=-c. This shows the translation of shifting left.

Horizontal shifts of the parent function y = log b ( x )

For any constant c , the function f ( x ) = log b ( x + c )

  • shifts the parent function y = log b ( x ) left c units if c > 0.
  • shifts the parent function y = log b ( x ) right c units if c < 0.
  • has the vertical asymptote x = c .
  • has domain ( c , ) .
  • has range ( , ) .

Given a logarithmic function with the form f ( x ) = log b ( x + c ) , graph the translation.

  1. Identify the horizontal shift:
    1. If c > 0 , shift the graph of f ( x ) = log b ( x ) left c units.
    2. If c < 0 , shift the graph of f ( x ) = log b ( x ) right c units.
  2. Draw the vertical asymptote x = c .
  3. Identify three key points from the parent function. Find new coordinates for the shifted functions by subtracting c from the x coordinate.
  4. Label the three points.
  5. The Domain is ( c , ) , the range is ( , ) , and the vertical asymptote is x = c .

Graphing a horizontal shift of the parent function y = log b ( x )

Sketch the horizontal shift f ( x ) = log 3 ( x 2 ) alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.

Since the function is f ( x ) = log 3 ( x 2 ) , we notice x + ( 2 ) = x 2.

Thus c = 2 , so c < 0. This means we will shift the function f ( x ) = log 3 ( x ) right 2 units.

The vertical asymptote is x = ( 2 ) or x = 2.

Consider the three key points from the parent function, ( 1 3 , −1 ) , ( 1 , 0 ) , and ( 3 , 1 ) .

The new coordinates are found by adding 2 to the x coordinates.

Label the points ( 7 3 , −1 ) , ( 3 , 0 ) , and ( 5 , 1 ) .

The domain is ( 2 , ) , the range is ( , ) , and the vertical asymptote is x = 2.

Graph of two functions. The parent function is y=log_3(x), with an asymptote at x=0 and labeled points at (1/3, -1), (1, 0), and (3, 1).The translation function f(x)=log_3(x-2) has an asymptote at x=2 and labeled points at (3, 0) and (5, 1).
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Questions & Answers

For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
Is there any rule we can use to get the nth term ?
Anwar Reply
how do you get the (1.4427)^t in the carp problem?
Gabrielle Reply
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
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Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
prince Reply
hello
Jessica Reply
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of ?105°F??105°F? occurs at 5PM and the average temperature for the day is ?85°F.??85°F.? Find the temperature, to the nearest degree, at 9AM.
Karlee Reply
if you have the amplitude and the period and the phase shift ho would you know where to start and where to end?
Jean Reply
rotation by 80 of (x^2/9)-(y^2/16)=1
Garrett Reply
thanks the domain is good but a i would like to get some other examples of how to find the range of a function
bashiir Reply
what is the standard form if the focus is at (0,2) ?
Lorejean Reply
a²=4
Roy Reply

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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