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Graphing Logarithmic Functions: Examples (page 2 of 3)

In the previous example, I said there were two options for how to graph logs. The previous page demonstrated how to work from the concept of logarithms to find nice neat points to plot. However, the other option is that you can use your calculator to find plot points.

To do this, you would use the change-of-
base formula to compute y = log2(x) by using y = ln(x)/ln(2):

  

y1 = ln(x) / ln(2)
  

  

...and get some plot points from your calculator's TABLE feature:

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TABLE for x = 1 to x = 6

TABLE for x = 7 to x = 12

TABLE for x = 13 to x = 18
  

Then adjust the initial value and the increment to get some plot points between x = 0 and x = 1:

  

TABLE for x = 1/8 to x = 7/8, counting by 1/8's

Depending upon your calculator's software, you will either get blank spaces for the y-values when x = 0 and when x is negative, or "ERROR" or "UNDEFINED" or some other error code, or else the program will crash. (Mine crashes for undefined y-values, which is why I was careful to start my TABLE display at a positive x-value.) In any case, this behavior in the TABLE feature reinforces the fact that logarithms are not defined for non-positive arguments.

(Regarding finding plot points between x = 0 and x = 1, if you do not know how to change your initial value from x = 0 or how to change your increment from 1, consult your owner's manual; the instructions will be somewhere in the chapter on graphing.)

If you are graphing the common (base-10) log or the natural (base-e) log, just use your calculator to get the plot points. In the case of the common log, if you tried to plot only whole-number points, then, in order to get as high as y = 2, you'd have to use x = 100, and your graph would be ridiculously wide. On the other hand, for the natural log, the base e is an irrational number anyway, so there's no point in trying to find nice neat plot points, because there aren't any (other than (1, 0), of course).

Sometimes the log graph is shifted a bit, either up, down, right, left, upside-down, or some combination of these. But the general shape tends to remain the same.

  • Graph y = log3(x) + 2.

    This is the basic log graph, but it's been shifted up by two units. To graph this, I will plug in useful values of x (powers of 3, either positive powers or negative powers) and simplify for the value of y.

      30 = 1, so log3(1) = 0, and log3(1) + 2 = 2
      31 = 3,       so log3(3) = 1, and log3(3) + 2 = 3
      32 = 9,       so log3(9) = 2, and log3(9) + 2 = 4
      33 = 27,       so log3(27) = 3, and log3(27) + 2 = 5

    Moving in the other direction:

      3–1 = 1/3, so log3( 1/3 ) = –1, and log3( 1/3 ) + 2 = 1
      3–2 = 1/9,       so log3( 1/9 ) = –2, and log3( 1/9 ) + 2 = 0
      3–3 = 1/27,       so log3( 1/27 ) = –3, and log3( 1/27 ) + 2 = –1

    These are the only "neat" points that I'm going to get for graphing. If I want further plot points, I can evaluate the function "ln(x) / ln(3)" in my calculator.

      

    The graph of y = log3(x) + 2 looks like this:

      

    graph of y = log_3(x) + 2 
     

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Cite this article as:

Stapel, Elizabeth. "Graphing Logarithmic Functions: Examples." Purplemath. Available from
    http://www.purplemath.com/modules/graphlog2.htm. Accessed
 

 

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