Learning to Use, and Love, Semilog Graph Paper

Learning to Use, and Love, Semilog Graph Paper
Remember, one of the main purposes of making graphs (besides torturing
students) is to find a straight line relationship between two quantities, so that you
can find the slope of your straight line and write, with confidence,
y-variable = (slope) * x-variable
In the graphs you have done so far, the axes were linear. What that means is, if an
axis went from 0 to 10, for example, then it was divided into ten equal parts, and it
was the same distance from 0 to 1 as from 5 to 6 as from 9 to 10. If an axis went
from 0 to 1 million, then it was again divided up into equal parts (maybe 5, maybe
10, maybe 20, depending on the size of your paper). The important thing was, it
was the same distance from 0 to 10 as from 1000 to 1010 or from 900,000 to
900,010. That’s what linear means.
The Limitations of Linear Graphs
Sometimes, however, linear axes aren’t much help. Consider the following
example. You have done an experiment where you took the following data:
x
y
-2 1.00
-1.5 1.30
-1 1.65
1 4.48
2.5 9.49
Now let’s graph these data on a traditional, linear graph.

10
9
8
7
6
5
4
3
2
1
0
-4 -3 -2 -1 0 1 2 3
Wow, that’s really not helpful. I wouldn’t want to guess at what kind of formula
that represents, but it’s sure not a straight line.
Now let’s try something. Take the base 10 logarithm of each of the y values and
fill them in to the third column of the table.
Then let’s make another plot, this time with log y on the vertical axis and x on the
horizontal axis.
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-4 -2 -0.2 0 2 4 6
-0.4

Bingo! A straight line! If we were so inclined, we could take that straight line and
find its slope.
Slope = (log y 2 – log y 1 )/(x 2 – x 1 )
And then you could write, with confidence,
log y = (slope) * x
as the formula for your graph.
This is not a difficult thing. But back in the days when dinosaurs ruled the earth,
there were no calculators. Logarithms had to be looked up in tables, a process just
fraught with opportunities for error, not to mention fatigue. So someone got the
bright idea, why don’t we figure out the logarithms ahead of time and mark their
distances on the paper to start with That way, we can plot the logs without
having to take those tedious logarithms all over again each time.
Thus was born semi-log graph paper. See an example of it below.

The horizontal axis is linear, as it was before. The vertical axis is now marked off
in UNEQUAL parts: they start off big and get smaller. (This is why it’s called
“semilog” paper, because one of the two axes is linear and the other one is log.
Graph paper that has both axes marked in logs is called “log-log” paper.)
Look at the vertical axis. If we say that the origin is 1, then the next major mark is
2, the next 3, and so on. Notice that the distance from 1 to 2 is not the same as
from 2 to 3. The axis is not linear. The distance from 1 to 2 represents the log of
2. The distance from 1 to 3 represents the log of 3, and so on like that. With this
paper, if the y value of your data point is 4, you don’t have to go find a logarithm,
you just mark your data point even with the 4 line, and its distance from the origin
will be proportional to the log of 4.
So, in order to graph that second data point listed above, you mark off your x-axis.
Then go to –1 on the x-axis, and then look at the y-axis. The number is 1.65, so

you obviously need a point between 1 and 2. If you look at the little lines between
1 and 2, you will see that there are 9 of them, not evenly spaced there either. The
first one represents 1.1, the second 1.2, the third 1.3, and on like that. Find 1.65
and mark the point there.
Now do the same for the rest of the data points. When you are done, we’ll look at
all the graphs and compare them.
Believe me, before there were calculators, this was one of the major time- and
labor-savers in scientific work.
A Small But Important Point
Where’s the zero on the y-axis Simple, there is none. Why not Because the log
of zero is UNDEFINED (dig up that little fact from your memory banks).
How Do We Get Equations From This I Thought That Was the Point
Now we take the slope, same as usual, except that these distances represent logs,
so the formula goes like this:
Slope = (log y 2 – log y 1 )/(x 2 – x 1 )
And we can say, with confidence, that log y = (slope) * x.
Now, if that is true, that means that y = 10 (slope)x . Whoopee.
If you have taken math for any length of time, you know that we would much
rather have formulas that take the form y = e x . The function e x is so simple to
work with that we would rather use it if we can. That means we have to get
natural (not base 10) logs in there somehow.
So what we do is, we find the slope as follows:
Slope = (ln y 2 – ln y 1 )/(x 2 – x 1 ) Call the slope a.
And then we can write, with confidence, y = e ax , which is a nice, easy formula.
Yes, you do have to find two natural logs there, but in the old days it was much
preferable to find two logs for each graph than to find one for every data point.

What if My Graph Doesn’t Go Through the Origin
Anytime you make a semilog graph, look at your graph and find the point on the
line where x = 0. If your y value equals 1, you’re home free, and y = e ax . If it
doesn’t, then let’s see how to fit that into the equation.
Suppose that at x = 0, y = A (that’s your y-intercept, for those who remember their
algebra). That means that ln y = ln A when x = 0.
What this boils down to is that you can then write
y = A e ax
Note for Those Who Want to Know Why
(And if You Don’t, You Can Skip This Part, BUT Remember the Formula)
ln y = ln A + (slope)*x
(recall y = mx + b from your math class)
the slope = a, so ln y = ln A + ax
(ln A +ax)
so y = e remember
y = (e ln A )(e ax )
your algebra!
but remember e ln A = A, so
y = A e ax
I Have a Better Idea; or, Why Deal With Base 10 Logs At All
Wouldn’t it just be easier to make log paper based on natural logs rather than base
10 logs Sure, but that’s not how it’s made. When you rule the world, you can
change all the graph paper to natural logs.
The Axes are Flexible, Too!
Another nice thing about this paper is that we can define the log axis as we need
to. Suppose, for example, that the data points went like this:

x
y
-2 1000
-1.5 1300
-1 1650
1 4480
2.5 9490
Here the y values range between 1000 and 10,000. So what we can do is say that
the origin is 1,000; the next major mark is 2,000; the next is 3,000; and so on up to
the top, which is 10,000. You can do that. Trust me. You can choose any range
you like, as long as the bottom value is some multiple of 10 and the top one is 10
times that.
So try it. Take another piece of semilog paper, mark the y-axis to go from 1000 to
10,000, and graph these data. How does your new graph compare with the other
semilog graph you drew
But What If My Data Extend Over More Than One Range
Or, The Magic of Multiple Cycles
Now, look again at your graph. Your y values were all between 1 and 10.
Suppose your data range from less than 1 to greater than 1000 Let’s look at
another set of data:
x
y
0 8325
1 4159
2 2081
3 1038
4 522
5 260
6 131
7 65
8 32
9 17
10 8

Now, our data go from less than 10 all the way up to a little less than 10,000.
What do you do
The piece of semilog paper we just used isn’t suitable. We can choose whether the
paper goes from 1 to 10, or 10,000 to 100,000, or .001 to .01, or whatever – but we
can’t make it go from 1 to 10,000. Not on this piece of paper.
You could go back to putting it on linear paper, and I invite the skeptics among
you to try it. But when the data are all spread out like that, it’s pretty difficult to
do accurate graphing on linear paper.
So what we do here is, we see how many CYCLES our data encompass. What
does that mean It has to do with powers of ten.
Our data go roughly from 1 to 10,000. Here are the cycles:
1 to 10 (our lowest number is between 1 and 10)
10 to 100
100 to 1000
1000 to 10,000 (our highest number is between 1000 and 10,000)
That will be sufficient for our data. That’s FOUR cycles (count them). The piece
of semilog paper we used before had only ONE cycle, but we need FOUR for
these data.
So we just take four pieces of semilog paper and we put them all together, and
then shrink them so they all fit on one standard piece of paper. See the example
below of FOUR-CYCLE semilog paper. You can have as many cycles as you
like, as long as it’s still legible.

And here’s how we mark the y-axis. The origin is 1, the next major mark is 10,
the next is 100, the next is 1000, and the next is 10,000. The marks between 1 and
10 are 2, 3, 4, and so on; between 100 and 1000 are 200, 300, 400, and so on. You
get the idea.
Now take the piece of 4-cycle semilog paper and graph these data. When you are
finished, we’ll look at all the graphs and compare them.
Congratulations! You are now an expert on how to use semilog paper. On the
final exam, you will have an opportunity to show off your exceptional knowledge
and skill.