Graphing Logs with a Calculator

How do you graph log functions with a calculator?

To graph a logarithmic function with a graphing calculator, such as the TI-84, follow these steps:

1. Always keep in mind that logs are inverses of exponentials; if you remind yourself of the expected shape of the graph, you'll better be able to see if an error was made in entering the function into your calculator.

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2. If the base of the log is something other than e or 10 (that is, if the log is something other than the natural log or the common log), then apply the change-of-base formula to convert the log expression into something your calculator will understand. For instance, if you are given y = log₃(x), use the formula to convert the function expression to .
3. Set the calculator's minimum for the x-values to be zero or, if your calculator's table utility is twitchy, to some value slightly greater than zero. (For instance, in my old TI-85, I have to set the minimum input value to be something like x = ½ to avoid the calculator freezing or crashing.)
4. Copy down the x-values and the corresponding y-values.
5. Plot the points as best you can, using one or two decimal places of accuracy.
6. Sketch in the line for the graph, taking care not to have the graph cross the vertical asymptote.

Note: If you're dealing with the natural log (that is, the base e log), then you'll be forced to use your calculator and its ln button, because the base e is not a nice, neat whole number. You'll be stuck with decimals, no matter what you do.

• Graph y = log₂(x)

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Depending upon your calculator's software, you will either get blank spaces in your table for the y-values when x = 0 and when x is negative; or the slots for those y-values will display "error", "undefined", or some other error code; or else the program will crash. (My TI-85 crashes for undefined y-values, which is why I was careful to start my TABLE display above at a positive x-value.)

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 table 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.)

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If you are graphing a common log (that is, the base-10 log) or a natural log (that is, the base-e log), just use your calculator to get the (approximate) plot points. All scientific and graphing calculators have dedicated buttons for these log bases.

When working with the common log, you will quickly reach awkwardly large numbers if you try to plot only whole-number points; for instance, in order to get as high as y = 2, you'd have to use x = 100, and your graph will be ridiculously wide (unless you change the horizontal scale, probably to a log scale).

When working with the natural log, the natural base e is an irrational number anyway, so there's no point in even trying to find nice neat plot points, because, other than the point (1, 0), there simply aren't any nice neat ones.

Sometimes the log graph is shifted a bit from the usual location (shown in the graph above); the graph can be shifted either up, down, right, left, or upside-down, or else some combination of these. But the general shape of the graph tends to remain the same.

• Graph y = log₃(x) + 2.

Note that this graph has the same shape as the previous graph; it's just been shifted two units upward.