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4.4 Graphs of logarithmic functions  (Page 6/8)

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

If  f ( x ) = − log b ( x ) If  f ( x ) = log b ( − x )
  1. Draw the vertical asymptote,   x = 0.
  1. Draw the vertical asymptote,   x = 0.
  1. Plot the x- intercept,   ( 1 , 0 ) .
  1. Plot the x- intercept,   ( 1 , 0 ) .
  1. Reflect the graph of the parent function   f ( x ) = log b ( x )   about the x -axis.
  1. Reflect the graph of the parent function   f ( x ) = log b ( x )   about the y -axis.
  1. Draw a smooth curve through the points.
  1. Draw a smooth curve through the points.
  1. State the domain,   ( 0 , ∞ ) , the range,   ( − ∞ , ∞ ) , and the vertical asymptote   x = 0.
  1. State the domain,   ( − ∞ , 0 ) , the range,   ( − ∞ , ∞ ) , and the vertical asymptote   x = 0.

Graphing a reflection of a logarithmic function

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

Before graphing   f ( x ) = log ( − x ) , identify the behavior and key points for the graph.

  • Since   b = 10   is greater than one, we know that the parent function is increasing. Since the input value is multiplied by   −1 , f   is a reflection of the parent graph about the y- axis. Thus,   f ( x ) = log ( − x )   will be decreasing as   x   moves from negative infinity to zero, and the right tail of the graph will approach the vertical asymptote   x = 0.  
  • The x -intercept is   ( −1 , 0 ) .
  • We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points.
Graph of two functions. The parent function is y=log(x), with an asymptote at x=0 and labeled points at (1, 0), and (10, 0).The translation function f(x)=log(-x) has an asymptote at x=0 and labeled points at (-1, 0) and (-10, 1).

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

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

Graph of f(x)=-log(-x) with an asymptote at x=0.

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

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Given a logarithmic equation, use a graphing calculator to approximate solutions.

  1. Press [Y=] . Enter the given logarithm equation or equations as Y 1 = and, if needed, Y 2 = .
  2. Press [GRAPH] to observe the graphs of the curves and use [WINDOW] to find an appropriate view of the graphs, including their point(s) of intersection.
  3. To find the value of   x , we compute the point of intersection. Press [2ND] then [CALC] . Select “intersect” and press [ENTER] three times. The point of intersection gives the value of   x , for the point(s) of intersection.

Approximating the solution of a logarithmic equation

Solve   4 ln ( x ) + 1 = − 2 ln ( x − 1 )   graphically. Round to the nearest thousandth.

Press [Y=] and enter   4 ln ( x ) + 1   next to Y 1 =. Then enter   − 2 ln ( x − 1 )   next to Y 2 = . For a window, use the values 0 to 5 for   x   and –10 to 10 for   y .   Press [GRAPH] . The graphs should intersect somewhere a little to right of   x = 1.

For a better approximation, press [2ND] then [CALC] . Select [5: intersect] and press [ENTER] three times. The x -coordinate of the point of intersection is displayed as 1.3385297. (Your answer may be different if you use a different window or use a different value for Guess? ) So, to the nearest thousandth,   x ≈ 1.339.

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Solve   5 log ( x + 2 ) = 4 − log ( x )   graphically. Round to the nearest thousandth.

x ≈ 3.049

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Summarizing translations of the logarithmic function

Now that we have worked with each type of translation for the logarithmic function, we can summarize each in [link] to arrive at the general equation for translating exponential functions.

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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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