1. Home
  2. College algebra
  3. Exponential and logarithmic
  4. Graphs of logarithmic functions

6.4 Graphs of logarithmic functions  (Page 3/8)

<< Chapter < Page
  College algebra     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

what does mean opportunity cost?
Aster Reply
what is poetive effect of population growth
Solomon Reply
what is inflation
Nasir Reply
what is demand
Eleni
what is economics
IMLAN Reply
economics theory describes individual behavior as the result of a process of optimization under constraints the objective to be reached being determined by
Kalkidan
Economics is a branch of social science that deal with How to wise use of resource ,s
Kassie
need
WARKISA
Economic Needs: In economics, needs are goods or services that are necessary for maintaining a certain standard of living. This includes things like healthcare, education, and transportation.
Kalkidan
What is demand and supply
EMPEROR Reply
deman means?
Alex
what is supply?
Alex
ex play supply?
Alex
Money market is a branch or segment of financial market where short-term debt instruments are traded upon. The instruments in this market includes Treasury bills, Bonds, Commercial Papers, Call money among other.
murana Reply
good
Kayode
what is money market
umar Reply
Examine the distinction between theory of comparative cost Advantage and theory of factor proportion
Fatima Reply
What is inflation
Bright Reply
a general and ongoing rise in the level of prices in an economy
AI-Robot
What are the factors that affect demand for a commodity
Florence Reply
price
Kenu
differentiate between demand and supply giving examples
Lambiv Reply
differentiated between demand and supply using examples
Lambiv
what is labour ?
Lambiv
how will I do?
Venny Reply
how is the graph works?I don't fully understand
Rezat Reply
information
Eliyee
devaluation
Eliyee
t
WARKISA
hi guys good evening to all
Lambiv
multiple choice question
Aster Reply
appreciation
Eliyee
explain perfect market
Lindiwe Reply
In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,
Ezea
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, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College algebra' conversation and receive update notifications?

Ask