1. Home
  2. Algebra and trigonometry
  3. Exponential and logarithmic
  4. Graphs of logarithmic functions

6.4 Graphs of logarithmic functions  (Page 3/8)

<< Chapter < Page
  Algebra and trigonometry     Page 3 / 8
Chapter >> Page >

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.

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

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.

Got questions? Get instant answers now!

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

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




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