Using lograrithmic graph paper

1. USING LOGRARITHMIC GRAPH
PAPER

2. Introduction
• Logarithmic graph paper is seldom seen by students in elementary
courses. There seems to be a general impression that (1) it is too difficult
to deal with, or (2) students "can pick it up" without instructions. Niether
is true, in my opinion.
• In the days of slide rules, students had (or ought to have had) intimate
familiarity with logrithms and logarithmic scales, for every slide rule had at
least two such scales. Nowadays many of these gory details are hidden in
the innards of an electronic calculator or computer, a "black box" that
grinds out numbers, whether or not those numbers have any significance.
• Graphs with logarithmic scales are found in research papers and
textbooks. One example is the usual graph of the electromagnetic
spectrum. If students are to interpret such graphs intelligently, they need
to directly experience the process of making one.
• All sorts of computer graphing software is available. The most used
software is designed for the needs of business, not science. Many such
software packages simply cannot do the things necessary for dealing with
the needs of physics.

3. LOGARITHMIC GRAPH PAPER
Logarithmic graph papers are
available in many types. They
simplify the process of
linearizing exponential and
power relations and
determining the constants in
their equation.

4. Logarithmic scales
• Starting from the left, we see labels from 1 to 10, repeated three times. Each segment labeled from 1 to 10 is called a
cycle. This sample has 3 cycles. The labeling already printed on the paper is for convenience only; the user must relabel the
axes to suit the data.
• For example, suppose you want to plot data values ranging from 2 units to 800 units on this three cycle log scale. The first
cycle is labeled "as is" from 1 to 9. The second cycle is relabeled 10, 20, 30, 40, 50, 60, 70, 80, and 90. The third cycle is
relabeled 100, 200, etc. to 1000.
• Each cycle covers a range of values spanning one factor of 10. The next cycle covers a range 10 times larger, etc. The value
zero does not appear on a logarithmic scale, for log (0) = minus infinity.
• The "generic" labeling is done differently on different brands of paper. Some label the start of a cycle with "1"; some use
"10". Some mark 1.5 and 2.5; some don't.
• The user has less freedom in labeling a log scale than a linear one. The scales are directed, i.e., one-way, and cannot be
labeled in the opposite direction.
• The underlying principle of the log scale is that lengths along the scale are proportional to the logarithms of the plotted
values. The C and D scales of slide rules are constructed the same way.
• The user must carefully examine the markings of the scale to correctly interpret its subdivisions. Some intervals have 10
subdivisions, some only five (every second one of the ten being shown). Errors in the use of log paper result from failure to
notice these differences in the way subdivisions are labeled.
• Graph papers with one linear and one logarithmic scale are called semi-logarithmic, log-linear, or simply logarithmic. Graph
papers with both scales logarithmic are called log-log, full log, or dual logarithmic.
• As is evident from the 10 divisions of each cycle, these logarithmic scales represent base-10 logarithms.

5. RESCALING LOGARITHMIC SCALE
We have so far only considered rescaling log scales by multiplying
each scale value by some factor of 10. This preserves the cycle
length. Other labelings are possible, but one must exercise great
care to avoid blunders in plotting.
(1) The scale values may be multiplied by any common factor. A
factor of 2 or 4 is easiest to deal with. This operation preserves
cycle length, but shifts the cycles along the axis. This can be useful
when the data spans only one factor of 10, but would fall in two
adjacent decades. See Fig. 7.11.
(2) The scale values may be raised to any common power. This
expands or contracts the cycle size. This can be useful when you
need two cycle paper but have only one cycle paper. Raise all of the
values printed along the scale to the second power. See Fig. 7.12.
This should be considered strictly an emergency expedient, never
acceptable in a graph intended for publication.
Another emergency trick is to splice the paper. Suppose you had
data that spanned only one factor of 10 but spanned two cycles of
the paper. For example, the values might range from 45 to 312. If
you used two cycle paper, this graph would occupy less than half of
the paper. If you cut one cycle paper at the "4" mark, it could be
spliced together so that it reads from 40 to 400.

6. SEMI-LOGARITHMIC GRAPH PAPER
Semilog graph paper is set up very differently from traditional
graph paper. While, on traditional graph paper, the y-axis
interval markings are at regular distances from one another
(for instance, 1 to 10, 10 to 20, 20 to 30), on semilog graph
paper, the y-axis is broken up into cycles, ranging from 0.1 to
1, 1 to 10, 10 to 100, and so forth. Within each cycle, the base
line is the beginning of the cycle, and each line above the base
line is the multiple of the line below it. For instance, in the 0.1
to 10 cycle, 0.1 would be the baseline, the next line would be
two times the baseline (0.2), the line after would be three
times the baseline (0.3), and so forth. Fortunately, making
graphs on semilog graphing paper is quite similar to making
graphs on regular graphing paper.

7. If you need logarithmic or any other paper, take
a look at these paper templates:
http://www.intmath.com/downloads/graph-
paper.php or here
http://www.papersnake.com/logarithmic/

8. Big thanks to
http://www.lhup.edu/~dsimanek/scenario/error
man/graphs2.htm
http://www.ehow.com/how_8397236_graph-
biology-data-semilog-paper.html