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.

Graph y
= log_{2}(x)

To
find my plot-points using my calculator (since my calculator can
only compute common, or base-10, and natural, or base-e, logs),
I will need to use the change-of-base
formula,
which gives me an equivalent equation. The original equation,
y
= log_{2}(x)
becomes y
=^{ln}^{(x)}/_{ln(2)}:

Once
I've entered the calculator-friendly form of the equation, I can
get some plot points from my calculator's TABLE feature:

(If
you have an old TI-85, so you have no in-built "TABLE"
utility, you can install any of the various after-market programs
which do much the same thing. The graphics to the right are screen
shots of what the
program
on my TI-85 produced.)

I
know I have to have all positive -values inside the log, so I
start the TABLE listing at x
= 1, and go from
there.

I'd
still like to have some plot points between zero and one, so I
adjust the initial ("start") value and the increment
(the "count by" amount) to get some additional plot
points between x
= 0 and x
= 1:

Using
these points (plotted to one or two decimal places in "accuracy"),
I will end up with the same graph as before.

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
slot will display "ERROR", "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 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 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. 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
would be ridiculously wide. When working with the natural log, the base
e
is an irrational number anyway, so there's no point in even trying
to find nice neat plot points, because, other than (1,
0), there aren't any.

Sometimes
the log graph is shifted a bit from the "usual" location (shown
in the graph above), 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
= log_{3}(x) + 2.

This is the basic log
graph, but it's been shifted
upward by two
units. To find plot points for this graph, I will plug in useful values
of x
(being powers of 3,
because of the base of the log) and then I'll simplify for the corresponding
values of y.

3^{0}
= 1, so
log_{3}(1) = 0, and
log_{3}(1)
+ 2 = 2
3^{1}
= 3, so
log_{3}(3) = 1, and
log_{3}(3)
+ 2 = 3
3^{2}
= 9, so
log_{3}(9) = 2, and
log_{3}(9)
+ 2 = 4
3^{3}
= 27, so
log_{3}(27) = 3, and
log_{3}(27)
+ 2 = 5

Moving in the other direction
(to get some y-values
for x
between 0 and
1):

3^{–1}
= ^{1}/_{3}, so
log_{3}(^{ 1}/_{3} ) = –1, and
log_{3}(^{
1}/_{3} ) + 2 = 1
3^{–2}
= ^{1}/_{9}, so
log_{3}(^{ 1}/_{9} ) = –2, and
log_{3}(^{
1}/_{9} ) + 2 = 0
3^{–3}
= ^{1}/_{27}, so
log_{3}(^{ 1}/_{27} ) = –3, and
log_{3}(^{
1}/_{27} ) + 2 = –1

These are the only "neat"
points that I'm going to bother finding for my graph. If I feel a need
for additional plot points, especially between any two of the points
I found above, I can evaluate the function "ln(x)
/ ln(3)"
in my calculator.