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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.

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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