How to Read Semi Log Graph Paper

7. Graphs on Logarithmic and Semi-Logarithmic Axes

by Yard. Bourne

In a semilogarithmic graph, one axis has a logarithmic calibration and the other centrality has a linear scale.

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

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

Y'all tin see some examples of semi-logarithmic graphs in this YouTube Traffic Rank graph.

Meet besides air force per unit area and Zipf Distributions subsequently on this page.

Semi-Logarithmic Graphs

In the following set of axes, the vertical scale is logarithmic (equal scale between powers of 10) and the horizontal scale is linear (even spaces between numbers).

There are no negative numbers on the y-axis, since we tin only find the logarithm of positive numbers.

Semilogarithmic axes.

Notation: The numbers on the y-axis become too close together nearly each integral power of 10, and then they have been removed for readability.

Instance 1: Graph of `y=x`

Let'south run across what the simple graph `y=x` looks like on different centrality types.

a. `y=ten` on Linear Axes

On ordinary linear axes, the graph of `y=x` is a straight line, passing through `(-2,-ii)`, `(-i,-1)`, `(0,0)`, `(ane,1)`, `(two,2)`, etc.

Graph of `y=x` on linear (lin-lin) axes.

b. `y=ten` on Semi-logarithmic Axes (vertical axis logarithmic, horizontal axis linear)

On semi-logarithmic axes, the graph of `y=x` is a curve, not a directly line. Information technology still passes through `(1,1)`, `(2,2)`, `(3,3)`, etc, but yous'll notice there are no negative values for `y` (and and so in this case, no negative values for `x` either) since we can't find the log of a negative number.

Graph of `y=x` on semilogarithmic (log-lin) axes.

I've marked the points `(1,1)`, `(2,ii)`, `(3,3)`, `(4,4)` `(v,5)`, `(6,6)` on the bend.

c. `y=ten` on Semi-logarithmic Axes (vertical centrality linar, horizontal centrality logarithmic)

Points along the curve `y=x` on lin-log axes.

I've marked the points `(1,1)`, `(2,2)`, `(3,3)` upwards to `(10,10)` on the curve.

Find there are no negative values for `x` on a lin-log bend.

Semi-logarithmic graph examples

(a) Traffic charts

The popularity of the site imeem.com grew very rapidly in 2006/7. Hither is Alexa's graph of that growth, using a linear horizontal scale (years) and a logarithmic verical scale for popularity rank (where rank=1 means nigh pop).

imeem was subsequently bought past MySpace.

Rank of imeem over time.

(b) Financial charts

The financial industry makes utilise of semi-logarithmic scales to make charts easier to read. See this Golden prices nautical chart as an case.

Instance 2: Variable Exponent

We conducted some observations in an experiment involving growth of a microbial population at unlike temperatures and obtained information equally follows:

T (°C)	−2	−1	0	1	2	3	four
P	0.020	0.143	one	7	49	343	2401

Nosotros observe the population increases by 7 times for each `i°` ascent in temperature, so nosotros can model the information using the role `P = seven^T`.

We plot the data on linear T-P axes as follows:

Graph of `P=seven^T` on linear-linear axes.

Our plot shows reasonable detail for values of `x` greater than `1`, merely doesn't tell united states of america much for smaller values of `10` or `y`. The points are too shut to the ten-axis for u.s.a. to meet what is going on.

Permit's now plot the same data on semi-logarithmic axes.

Graph of `P=seven^T` on semilogarithmic axes.

Now we can see a lot more than information for smaller values of `10` and `y`. This is the dazzler of semi-logarithmic axis plots - you can run into more item in graphs where at that place is a very wide range of values, merely some of the information is close together.

Example iii

Draw the graphs of `y=five^x` on linear and both types of semilogarithmic axes.

Answer

Log-log Graphs

Log-log graphs utilise a logarithmic scale for both vertical and horizontal axes.

Here's the graph of `y=10` on log-log axes.

ten^-2

10^-1

10⁰

10^ane

10ⁱⁱ

Points forth the line `y=10` on log-log axes.

The above graph has the following points highlighted for clarity:

`(1, 1)`, `(two, 2)`, `(three, 3)`, `(four, 4)`, `(5, v)`, `(half dozen, 6)`, `(x, 10)`, and `(100, 100)`.

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

Population Instance

Nosotros tin can as well graph the population example we met earlier, `P = seven^T` on log-log newspaper (i.e. both axes employ log scales):

10^-ane

10⁰

xⁱ

x²

10^three

10^iv

Points along the bend `P=7^T` on log-log axes.

We tin see even more detail for small values of x and y at present, however we tin't include the negative temperatures on the horizontal axis.

Instance 4: Variable Raised to a Fractional Exponent

Graph y = 10 ^1/2 using all four axis types (rectangular, both types of semi-log, and log-log). This function is equivalent to `y=sqrt(x)`.

Answer

Application 1: Air pressure level

i. By pumping, the air force per unit area in a tank is reduced by 18% each second. So the percentage of air force per unit area remaining at time `t` 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.

Answer

Application ii: Zipf Distributions

Consider the most mutual words in English. It turns out that at that place is a human relationship between the rank of a discussion'due south occurrence and the frequency of its use. That human 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 `one/grand`. In other words, the 2nd most commonly used give-and-take occurs about `i/two` as oftentimes as the nearly common word. Too, the 3rd about common discussion occurs about `1/iii` every bit oftentimes as the most common word.

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

• Calls to estimator operating systems
• Colors in images
• Equally the basis of most approaches to prototype compression
• City populations (a pocket-size number of large cities, a larger number of smaller cities)
• Wealth distribution (a small number of people have large amounts of coin, large numbers of people accept 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.

a. Mutual 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 tabular array is based on the Brown Corpus, a careful study of a one thousand thousand words from a wide variety of sources including newspapers, books, magazines, fiction, government documents, comedy and academic publications.

The nigh common discussion, "the" occurred around `70,000` times (or `vii%` of the one thousand thousand words counted). The next ranked word, "of", occurred around `3.6%` of the time (or well-nigh `1/2` as oftentimes as the pinnacle-ranked discussion.) The third nigh popular discussion was "and", with a frequency of `two.eight%`, or roughly `1/iii` of the frequency of the acme ranked word.

Rank	Word	Frequency	% Frequency	Theoretical Zipf Distribution
ane	the	69970	6.8872	69970
2	of	36410	iii.5839	36470
three	and	28854	ii.8401	24912
4	to	26154	two.5744	19009
five	a	23363	2.2996	15412
half-dozen	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
xi	for	9489	0.9340	7345
12	information technology	8760	0.8623	6768
13	with	7290	0.7176	6277
xiv	as	7251	0.7137	5855
15	his	6996	0.6886	5487
sixteen	on	6742	0.6636	5164
17	be	6376	0.6276	4878
18	at	5377	0.5293	4623
19	by	5307	0.5224	4394
xx	I	5180	0.5099	4187

(The first 20 words in the Brownish Corpus, published in 1967. This Corpus is the count of how often one meg words were used in a variety of books, newspapers and other publications. [Tabular array source no longer bachelor, but like to Corpus of Gimmicky American English.]

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

Permit's plot what we accept observed:

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

`f/(1^0.94) = 69970`,

`f/ii^0.94 = 69970/2^0.94 = 36470, `

`f/3^0.94 = 69970/3^0.94 = 24912,`

`f/iv^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. (I but did trial and mistake in Excel until I plant the closes fit.)

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

We now plot the pinnacle 2000 English words and use a log-log scale (log of the rank for the horizontal centrality 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.

We come across that there is a remarkably consistent result for the summit 2000 most-used English words. For your information, the final few in the list of 2000 words are:

          1992nd	device       1993rd	comport       1994th	runs       1995th	improved        1996th	games       1997th	cultural       1998th	plenty       1999th	mile       2000th	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 a sample of 500,000 page views in Interactive Mathematics, the nearly commonly visited folio was the homepage, with 27,855 views. The next virtually common page was the Algebra Introduction, with around 1/2 of the views. The tertiary ranked folio had well-nigh 1/three of the views of the nearly popular page.

Rank	Page	Frequency (Folio views)
1	Domicile	`27855`
2	Basic Algebra Introduction	`15334`
three	Addition & Subtraction in Algebra	`7605`
4	Math Of Beauty	`5965`
5	Graphs of Sine and Cosine	`5749`
half dozen	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 summit 500 pages in the site, we take the following log-log graph of the page views:

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

`27855/2^0.67 = 17507`

`27855/3^0.67 = 13342`

`27855/iv^0.67 = 11003`

`27855/5^0.67 = 9475`

`27855/6^0.67 = 8835 `

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

Encounter also Zipf Distributions, log-log graphs and Site Statistics over in the IntMath blog.

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Source: https://www.intmath.com/exponential-logarithmic-functions/7-graphs-log-semilog.php