Logarithms and orders of magnitude

Logarithms and orders of magnitude are useful concepts for handling numbers that span a very wide range. Logarithms help us understand how numbers grow, simplify complex calculations and can help to display data sets meaningfully in the form of a graph when there is a very wide variation in the measured quantities.

Use this page to revise the following concepts within computation and magnitude:

Orders of magnitude

An order of magnitude describes the approximate scale of a number as a power of \(10\), helping to compare values based on scale or size rather than exact amount, by comparing it to powers of \(1\). For example

\(10\) is one order of magnitude greater than \(1\) (because \(10^1=10\))

\(1,000\) is three orders of magnitude greater than \(1\) (because \(10^3=1000\))

\(0.01\) is two orders of magnitude smaller than \(1\) (because \(10^{-2}=0.01\))

Orders of magnitude are often used when comparing very large or very small values. For example,  the mass of the Earth, \(~10^{24} \text{ kg}\) is about \(24\) orders of magnitude larger than \(1\text{ kg}\).

Logarithms

A logarithm is an extension of the idea of orders of magnitude that allows you to precisely compare numbers that may differ by orders of magnitude.

A logarithm tells you what power you need to raise a given base (usually \(10\)) to in order to get a certain number - for example, \( \log _{10} (1000) = 3\) because \(10^3 = 1000\).

Some common applications that employ logarithmic scales include the Richter scale for measuring earthquake strength, the decibel scale for measuring the loudness of sound, and the pH scale for measuring the strength of acids and bases. These convert very large or small numbers to a scale that is more useful. A change in pH of \(3\) refers to a change of \(3\) orders of magnitude, or \(10^3=1000\). Therefore according to the scale depicted below, stomach acid is \(1000\) times more acidic than coffee.

A visual representation of the pH scale, with notable substances for each pH. For example, coffee at approximately pH 6 and stomach acid at approximately pH 2

Logarithms and displaying Data

Logarithms are especially useful in the display of data where quantities may differ by orders of magnitude. If most of the data points are within a fairly narrow range but a few are much larger or smaller, the width of the overall range will tend to obscure the detail within the majority range.

Consider data consisting of annual incomes. It may be that many of the incomes recorded are less than \(\$100,000\) but a few are more than \(\$10,000,000\).

Suppose the following numbers represent costs of items:

ItemCost ($)Visualisation
A 10,000 A graph representing logarithms and orders of magnitude. Along the vertical (value) axis are numbers between 0 and 14,000,000, represented at intervals of 2,000,000. Along the horizontal (category) axis are the letters A, B, C, D, E, F, G and H. At the letter C on the horizontal axis a blue line reaches half-way between 0 and 2,000,000 on the vertical axis. At the letter F on the horizontal axis a blue line reaches a quarter of the way between 0 and 2,000,000 on the vertical axis. At the letter G on the horizontal axis a blue line rises just above the 2,000,000 marker on the vertical axis. At the letter H on the horizontal axis, a blue line rises to the 12,000,000 marker on the vertical axis.
B 60,000
C 1,200,000
D 20,000
E 79,000
F 750,000
G 2,500,000
H 12,000,000

Notice that there is quite a difference between the size of the largest and smallest numbers in this set. Any difference between smaller values (A, B, D and E) is visually indistinguishable.

If the base \(10\) logarithm of each number is taken, the following set is obtained:

ItemCost ($)Visualisation
A 4 A graph representing logarithms and orders of magnitude titled Item cost ($) (log_10 scale). Along the vertical (value) axis (labelled Cost ($; log_10)) are numbers between 0 and 8, represented at intervals of 1. Along the horizontal (category) axis (labelled items) are the letters A, B, C, D, E, F, G and H. At the letter A a blue line reaches to the number 4. At the letter B a blue line reaches almost to the number 5. At the letter C a blue line reaches to the number 6. At the letter D a blue line reaches to just past the number 4. At the letter E a blue line reaches to just under the number 5. At the letter F a blue line reaches to just under the number 6. At the letter G a blue line reaches to just above the number 6. At the letter H a blue line rises to the number 7
B 4.778  
C 6.079  
D 4.301  
E 4.898  
F 5.875  
G 6.398  
H 7.079  

These values are much closer together and easier to interpret. This demonstrates why logarithmic scales are so useful.

Worked Example

Consider the following populations of towns and cities in Victoria.

Town/CityPopulation
Melbourne 3 707 530
Geelong 143 921
Ballarat 85 000
Horsham 15 292
Churchill 4 750
Willcannia 688
Yanac 84

Graph this data on a histogram.

Step 1 - Convert each value as a value \(\log_{10}\) value.

Town/CityPopulationLog(population)
Melbourne 3 707 530 6.57
Geelong 143 921 5.16
Ballarat 85 000 4.93
Horsham 15 292 4.18
Churchill 4 750 3.68
Willcannia 688 2.84
Yanac 84 1.92

Step 2 - Create a histogram.

A graph representing logarithms and orders of magnitude titled population distribution (log_10 scale). Along the vertical (value) axis (labelled population (log_10)) are numbers between 0 and 8, represented at intervals of 1. Along the horizontal (category) axis (labelled town/city) are the names Melbourne, Geelong, Ballarat, Horsham, Churchill, Wilcannia and Yanac. At the name Melbourne a blue line reaches to two thirds of the way to number 7. At the name Geelong a blue line reaches to just beyond the number 5. At the name Ballarat a blue line reaches to just before the number 5. At the name Horsham a blue line reaches to just past the number 4. At the name Churchill a blue line reaches to between the numbers 3 and 4. At the name Wilcannia a blue line reaches to just under the number 3. At the name Yanac a blue line reaches to just under the number 2.