GRAPHING WITH LOGARITHMIC PAPER
© Department of Physics, University of Guelph
Guelph, Ontario, Canada
Go to this site to print free graph paper (en francais aussi): http://www.printfreegraphpaper.com/
||At the end of the
on Graphing Simple Functions, you saw
to produce a linear graph of the exponential function N = N0
eat as shown in panel 1. This was done by taking the natural
logarithm of both sides of the equation and plotting ln(N/N0)
vs t to get a straight line of slope a.
||Sometimes it's a
to look up a bunch of logarithms of values of N/N0 so we
use of a special type of graph paper which does this automatically. A
of this paper is shown in panel 2. Notice that it has a linear scale
but a logarithmic scale vertically. It's called
Notice that the vertical scale goes from 1 to 10. This paper is called
"one-cycle semi-logarithmic paper". The significance of
will become apparent in a little while.
||In panel 3,
there's a table
of values of N/N0 which obey an exponential relationship. In
the right-hand column, I've looked up the natural logarithms of N/N0
||The graph shown in panel 4 is a plot of these
t. Notice that this graph is on normal graph paper, not semi-log paper.
We'll use semi-log paper in a moment. As you see, the graph is a
line and its slope, and thus the constant a, can be found. Pause for a
moment and check the calculation of a.
||Now let's see how the semi-log paper simplifies
as shown in panel 5. The same data as in panel 3 is used here and,
our N/N0 data is all between 1 and 10, we can use the
on the left-hand edge of the graph paper just as they stand. All we
to do is plot the numbers as given. We don't have to find logarithms,
paper does it for us. That's the beauty of semi-log paper. You have to
watch out how the paper is sub-divided, though. In this example, it's
in 0.1, from 1 to 3, but in divisions of 0.2, from 3 to 5 and 0.5 from
5 to 10. Pause here and see how this graph is plotted.
Now let's find a. Again, we must find the slope and this will
finding logarithms but only at two points on whatever triangle we use
determine the slope. Pause again and check the calculation of the slope
in panel 5.
Notice, in fact, we had to look up only one logarithm in the
when we remembered that the difference of two logs is the log of the
Of course, we got the same value as before,
a = 0.23 s-1 but with a lot less work.
||Suppose, however, our data had been as shown in
of panel 6. Now the values of t are the same but the values of N/N0
are 10 times larger. What do we do now? The answer is that the decade
which the vertical axis runs is quite arbitrary. It can be 1 to 10 as
or it can be 10 to 100 which is what we need now, or 100 to 1000, or
to 1, and so on. Pause and see that you understand how this graph in
6 was plotted.
||Now let's suppose that you have the data given
in the table
on panel 7. None of the semi-log paper you have seen up to this point
work. You could plot the first number, or the 2nd to 5th, or the 5th to
7th, but you couldn't plot them all. Your one-cycle paper will go only
from 1 to 10, or 10 to 100, or 100 to 1000, in other words, one decade.
But now N/N0 goes over parts of 3 decades, that is 1 to
For this you need three-cycle semi-log paper which has been
here to plot this data. Pause and check over the plot and calculation
As you can see from this, you choose the number of cycles in
paper you use to match the span of data which you have; semi-log paper
comes in one, two, three, seven cycles etc.
||Let's now turn to
problem. Suppose you were presented with the set of data shown in panel
8. A graph of y vs x is also shown in panel 8, and you can see it's a
curve. But other than that, it's not very informative. Suppose,
in addition, there were theoretical grounds for believing that this
obeyed a power-law, y = axb. How could we find if this were
true and, if it were, evaluate the constants a and b?
||Let's take logarithms of both sides of the
equation as in
panel 9. You can see that, if y = axb, then a graph of log y
vs log x yields a straight line of slope b and y intercept log a.
if a graph of log y vs log x for a set of data is a straight line, then
the data does indeed follow the relation y = axb. Now we
look up a table of values of log x and log y and plot it but I won't
to do it because, just as with the exponential law, there's a simpler
In this case, the y intercept is log 2.5. Remember that it is not just
2.5 since the vertical axis is logarithmic, so we now have log a = log
2.5, so a must be 2.5. Therefore, we can write that this data fits the
equation y = 2.5x0.41. You should really include the proper
units with the value of a. To find them, simply rearrange the equation
y = a x 0.41 to solve for a, in other words, a = y/x0.41
and note that the units for both y and x were given as metres. So the
for a must be metres0.53. One final point about this graph.
Suppose the horizontal axis didn't start at 1, and there's no need that
it should. After all, your values of x might have been between 10 and
so you would have started your horizontal axis at 10. In that case, you
couldn't read the y intercept right off the graph. It must be read
log x = 0, in other words, where x = 1. To find the value of a, in this
case, you just use the equation y = a xb, substitute values
for x, y and b, and solve for a.
||Since we must plot log y vs log x, we need graph
logarithmically along both axes. It's called
"log-log paper" and
a 1 x 1 cycle sample is shown in panel 10 where our data of panel 8 is
plotted. Pause and see that you understand how the points were plotted.
The graph is a straight line so the data does obey y = axb.
Now let's find a and b. The constant b is given by the slope. Study
11 and see that you understand how the value was obtained. In
the slope, you may use either logs to the base 10 or logs to the base e
as long as you are consistent. Notice that since logs have no units,
the slope has no units.
The value of log a is the same as the value of the y
intercept. To obtain
this, we look on the graph for the point where the horizontal variable
is 0. Remember that since the horizontal axis is logarithmic, the
variable is actually log x, not just x, so we want the point where log
x = 0. In order for log x to be 0, x must be 1. The y intercept can be
read off the graph along the vertical line where x = 1.
If values of x and y extend over more than one decade, then more
must be used. Log-log paper comes in many combinations, such as 2 x 1,
2 x 3 and 5 x 3.