7. Graphs on Logarithmic and Semi-Logarithmic Paper

by M. Bourne

In a semilogarithmic graph, one axis has a logarithmic scale and the other axis has a linear scale. You can see some examples of semi-logarithmic graphs in this YouTube Traffic Rank graph and in this article on loudspeakers (external site). See also air pressure and Zipf Distributions later on this page.

Need Graph Paper?

Download graph paper, including
log-log and semi-log.

In log-log graphs, both axes have a logarithmic scale.

The idea here is we use semilog or log-log graph paper so that we can more easily see details for small values of y as well as large values of y.

Semi-Logarithmic Graphs

Example 1: Variable Exponent

Plot the graph of y = 5^x on normal and then semilogarithmic paper.

Answer:

We first graph y = 5^xusing ordinary x- and y- linear scales (the space between each unit remains fixed for both axes):

We see that the detail for anything less than x = 2 is lost.

Using a semilogarithmic scale on the y axis gives:

math expression

We can now see much more detail in the y values when x < 2.

Notice that the numbers along the x axis are evenly spaced, while along the y-axis, we have powers of 10 evenly spaced.

What does the scale mean on the y-axis? To see more clearly, let's zoom in on the section between 1 and 10:

Now let's zoom in on the section between 0.1 and 1.0.

Log-log Graphs

We can also graph y = 5^x on log-log paper (i.e. both axes use log scales)

NOTE: Both the domain (x-values) and the range (y-values) must be POSITIVE, because you cannot have the logarithm of a negative number.

math expression

We can see even more detail for small values of x and y now.

Let's have a play. I have drawn an exponential curve for you in LiveMath. You can see what it looks like on a semilogarithmic scale; and on a log-log scale. Change the function to anything you like (within reason) and see what it looks like in semi-log and log-log format.

LIVEMath

Example 2: Variable Raised to a Fractional Exponent

Let's now graph y = x^1/2 using all 3 axis types. This function is equivalent to y = √x.

Using rectangular axes, we can see that the graph of y = x^1/2 is half of a parabola on its side (i.e. its axis is vertical):

sqrt x

We have seen this curve before, in The Parabola section.

Note 1: The detail near (0, 0) is not so good using a rectangular grid.

Note 2: The curve passes through (0, 0), (1, 1), (4, 2) and (9, 3). In each case, the y-value is the square root of the x-value, which is to be expected.

Let's see the curve using a semi-logarithmic plot.

sqrt x semi-log

Now we have a lot better detail for small x. The lowest value of y that the graph indicates is y = 0.1. We cannot show y = 0, since the logarithm of 0 is not defined.

We can see that the curve still passes through (1, 1), (4, 2) and (9, 3).

And now for the log-log graph:

sqrt x log-log

We observe that the graph of y = √x is a straight line when graphed on log-log axes.

Our curve passes through (1, 1), (4, 2) and (9, 3), as it should.

Application 1: Air pressure

1. By pumping, the air pressure in a tank is reduced by 18% each second. So the percentage of air pressure remaining is given by p = 100(0.82)^t.

Plot p against t for 0 < t < 30 s on

(a) a rectangular co-ordinate system

(b) a semilogarithmic system.

Try it on paper first, and then see what you get using the LiveMath example above.

The answer is given below.

Answer:

(a) Rectangular plot:

math expression

(b) Semilogarithmic Plot:

math expression

Application 2: Zipf Distributions

Consider the most common words in English. It turns out that there is a relationship between the rank of a word's occurrence and the frequency of its use. That relationship was observed by George Kingsley Zipf in the first half of the 20th century.

The Zipf Distribution is an observation comparing rank and frequency of word occurrences. In general, the word with rank k has a frequency roughly proportional to 1/k. In other words, the second most commonly used word occurs about 1/2 as often as the most common word. Likewise, the 3rd most common word occurs about 1/3 as often as the most common word.

Zipf Distributions occur naturally in many situations, for example in:

Calls to computer operating systems
Colors in images
As the basis of most approaches to image compression
City populations (a small number of large cities, a larger number of smaller cities)
Wealth distribution (a small number of people have large amounts of money, large numbers of people have small amounts of money)
Company size distribution
Artificial intelligence (in particular, "chat bots" that can chat with humans) relies on the limited number of questions and statements that people actually write in chats. (See ALICE).

a. Common English Words

Zipf originally developed his law in response to the observation that the frequency of words was inversely proportional to the rank of each word.

For example, the most common 20 words in English are listed in the following table. The table is based on the Brown Corpus, a careful study of a million words from a wide variety of sources including newspapers, books, magazines, fiction, government documents, comedy and academic publications.

The most common word, "the" occurred around 70,000 times (or 7% of the million words counted). The next ranked word, "of", occurred around 3.6% of the time (or about 1/2 as often as the top-ranked word.) The third most popular word was "and", with a frequency of 2.8%, or roughly 1/3 of the frequency of the top ranked word.

Rank	Word	Frequency	% Frequency	Theoretical Zipf Distribution
1	the	69970	6.8872	69970
2	of	36410	3.5839	36470
3	and	28854	2.8401	24912
4	to	26154	2.5744	19009
5	a	23363	2.2996	15412
6	in	21345	2.1010	12985
7	that	10594	1.0428	11233
8	is	10102	0.9943	9908
9	was	9815	0.9661	8870
10	he	9542	0.9392	8033
11	for	9489	0.9340	7345
12	it	8760	0.8623	6768
13	with	7290	0.7176	6277
14	as	7251	0.7137	5855
15	his	6996	0.6886	5487
16	on	6742	0.6636	5164
17	be	6376	0.6276	4878
18	at	5377	0.5293	4623
19	by	5307	0.5224	4394
20	I	5180	0.5099	4187

(The first 20 words in the Brown Corpus, published in 1967. This Corpus is the count of how often one million words were used in a variety of books, newspapers and other publications. Table source.

I have included the "Theoretical Zipf Distribution, based on the n-th ranked word occurring approximately 1/n times the frequency of the highest ranked word. This gives us a hyperbola, that we met before.)

Let's plot what we have observed:

brown corpus

The dark blue data points represent the top 20 occurring English words (with the first few labeled). The pink line is the theoretical Zipf distribution, which is found to be f/(n^0.94), where f is the frequency of the top-ranked word and n is the rank of the word.

f/(1^0.94) = 69970,
f/2^0.94 = 69970/2^0.94 = 36470,
f/3^0.94 = 69970/3^0.94 = 24912,
f/4^0.94 = 69970/4^0.94 = 19009,
f/5^0.94 = 69970/5^0.94 = 15412,
...

The power 0.94 comes from observing the best line of fit for the word frequencies.

There is a fairly large gap in the pattern for the words "to", "a" and "in", but it settles down and is quite consistent after that.

We now plot the top 2000 English words and use a log-log scale (log of the rank for the horizontal axis and log of the frequency for the vertical axis). If a distribution gives us a straight line on a log-log scale, then we can say that it is a Zipf Distribution.

brown corpus log-log

We see that there is a remarkably consistent result for the top 2000 most-used English words. For your information, the last few in the list of 2000 words are:

1992^nddevice
1993^rd conduct
1994^th runs
1995^th improved
1996^th games
1997^th cultural
1998^th plenty
1999^th mile
2000^th components

b. Websites and the Zipf Distribution

We also observe a Zipf Distribution when it comes to popularity of pages in Websites.

For example, out of the most recent 500,000 page views in Interactive Mathematics, the most commonly visited page is the homepage, with 27,855 views. The next most common page is the Algebra Introduction, with around 1/2 of the views. The 3rd ranked page has about 1/3 of the views of the most popular page.

Rank	Page	Frequency (Page views)
1	Home	27855
2	Basic Algebra Introduction	15334
3	Addition & Subtraction in Algebra	7605
4	Math Of Beauty	5965
5	Graphs of Sine and Cosine	5749
6	Volume of Solid of Revolution	5667
7	Trigonometric Graphs Introduction	5584
8	Download LiveMath	5517
9	Introduction to Trigonometric Functions	4701
10	Sitemap	4309

For the top 500 pages in the site, we have the following log-log graph of the page views:

intmath - zipf distribution

The theoretical Zipf Distribution (the pink line) is obtained as follows. The power used, 0.67, once again comes from observing the best line of fit.

27855/2^0.67 = 17,507
27855/3^0.67 = 13,342
27855/4^0.67 = 11,003
27855/5^0.67 = 9,475
27855/6^0.67 = 8,835

After the page ranked 200th, the pattern breaks down, but interestingly, from the 300th to the 500th page, there is still a consistent relationship between rank and frequency.

See also Zipf Distributions, log-log graphs and Site Statistics over in the math blog.