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6.4 Graphs of logarithmic functions  (Page 5/8)

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Given a logarithmic function with the form   f ( x ) = a log b ( x ) , a > 0 , graph the translation.

  1. Identify the vertical stretch or compressions:
    • If   | a | > 1 , the graph of   f ( x ) = log b ( x )   is stretched by a factor of   a   units.
    • If   | a | < 1 , the graph of   f ( x ) = log b ( x )   is compressed by a factor of   a   units.
  2. Draw the vertical asymptote   x = 0.
  3. Identify three key points from the parent function. Find new coordinates for the shifted functions by multiplying the   y   coordinates by   a .
  4. Label the three points.
  5. The domain is   ( 0 , ∞ ) , the range is   ( − ∞ , ∞ ) , and the vertical asymptote is   x = 0.

Graphing a stretch or compression of the parent function y = log b ( x )

Sketch a graph of   f ( x ) = 2 log 4 ( x )   alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

Since the function is   f ( x ) = 2 log 4 ( x ) , we will notice   a = 2.

This means we will stretch the function   f ( x ) = log 4 ( x )   by a factor of 2.

The vertical asymptote is   x = 0.

Consider the three key points from the parent function,   ( 1 4 , −1 ) , ( 1 , 0 ),   and   ( 4 , 1 ) .

The new coordinates are found by multiplying the   y   coordinates by 2.

Label the points   ( 1 4 , −2 ) , ( 1 , 0 )   , and   ( 4 , 2 ) .

The domain is   ( 0,   ∞ ) , the range is   ( − ∞ , ∞ ),   and the vertical asymptote is   x = 0.   See [link] .

Graph of two functions. The parent function is y=log_4(x), with an asymptote at x=0 and labeled points at (1, 0), and (4, 1).The translation function f(x)=2log_4(x) has an asymptote at x=0 and labeled points at (1, 0) and (2, 1).

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

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Sketch a graph of   f ( x ) = 1 2   log 4 ( x )   alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

Graph of two functions. The parent function is y=log_4(x), with an asymptote at x=0 and labeled points at (1, 0), and (4, 1).The translation function f(x)=(1/2)log_4(x) has an asymptote at x=0 and labeled points at (1, 0) and (16, 1).

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

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Combining a shift and a stretch

Sketch a graph of   f ( x ) = 5 log ( x + 2 ) .   State the domain, range, and asymptote.

Remember: what happens inside parentheses happens first. First, we move the graph left 2 units, then stretch the function vertically by a factor of 5, as in [link] . The vertical asymptote will be shifted to   x = −2.   The x -intercept will be   ( −1, 0 ) .   The domain will be   ( −2 , ∞ ) .   Two points will help give the shape of the graph:   ( −1 , 0 )   and   ( 8 , 5 ).   We chose   x = 8   as the x -coordinate of one point to graph because when   x = 8,     x + 2 = 10,   the base of the common logarithm.

Graph of three functions. The parent function is y=log(x), with an asymptote at x=0. The first translation function y=5log(x+2) has an asymptote at x=-2. The second translation function y=log(x+2) has an asymptote at x=-2.

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

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Sketch a graph of the function   f ( x ) = 3 log ( x − 2 ) + 1.   State the domain, range, and asymptote.

Graph of f(x)=3log(x-2)+1 with an asymptote at x=2.

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

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Graphing reflections of f ( x ) = log b ( x )

When the parent function   f ( x ) = 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. To visualize reflections, we restrict   b > 1,   and observe the general graph of the parent function   f ( x ) = log b ( x )   alongside the reflection about the x -axis,   g ( x ) = −log b ( x )   and the reflection about the y -axis,   h ( x ) = log b ( − x ) .

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) when b>1 is the translation function with an asymptote at x=0. The graph note the intersection of the two lines at (1, 0). This shows the translation of a reflection about the x-axis.

Reflections of the parent function y = log b ( x )

The function   f ( x ) = −log b ( x )

  • reflects the parent function   y = log b ( x )   about the x -axis.
  • has domain,   ( 0 , ∞ ) , range,   ( − ∞ , ∞ ) , and vertical asymptote,   x = 0 , which are unchanged from the parent function.


The function   f ( x ) = log b ( − x )

  • reflects the parent function   y = log b ( x )   about the y -axis.
  • has domain   ( − ∞ , 0 ) .
  • has range,   ( − ∞ , ∞ ) , and vertical asymptote,   x = 0 , which are unchanged from the parent function.

Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
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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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