Logarithms

Introduction

Logarithms were invented by Napier. Before we had calculators, logarithms made calculation easier because they reduced multiplication and divisions to addition and subtraction. These days, logarithms are less important for this purpose.

Logarithms

A logarithm is writen as log_a(x) where a is a number called the base. Usually logarithms are written to the base 10 or sometimes base 2 for binary numbers, but it can be any number. If the logarithm is to the base e= 2.71828..., then we call it a natural logarithm because it is the only logarithm base which has a rate of change equal to the thing which is changing. Natural logs are also written, ln(u). The ln comes from the Latin, log naturalis.

Rules of Logarithms

log_a(0) is undefined         (1)
log_a(1) = 0                          (2)
log_a(u v) = log_a(u) + log_a(v)        (3)
log_a(u/v) = log_a(u) - log_a(v)         (4)
log_a(u)ⁿ = n log_a(u)                      (5)
log_b(u) = log_a(u)/log_a(b) - change of base      (6)
Where b is the old base, a is the new base, u is the argument of the logarithm.
log_a(1/u) = - log_a(u) from (2) and (4).                (7)
With these rules we can manipulate the exponential functions. Logs are the inverse functions of

Exponential Function

We have seen how to convert a number into a logarithmic number but how about if we are given a logarithm of a number and want to know what the original number was? This is achieved using exponential functions. The exponential function is the inverse function of a logarithmic function.
That is to say, that a^log_a(u) = u or log_a(a^u)=u
If we have a log to the base 10, the inverse function is 10^u.
For natural logarithms, the inverse function is e^u or exp(u). In particular, log₁₀(10^u) = u
and 10^log₁₀(u)= u (8)
ln(e^u) = u or e^ln(u) = u (9)

Example

What is the value of the number the gives the following logs to the base 10:
i) 1, ii) 12, iii) -2, iv) 2.6, v) -5.43

Answers

log₁₀ x = 1
x=10¹ = 10
log₁₀ x = 12
x=10¹² = 1x10¹²
log₁₀ x = -2
x=10^-2 = 0.02
log₁₀ x = 2.6
x=10^2.6 = 501.187
log₁₀ x = 12
x=10^-5.43 = 3.71535 x 10^-6

Logarithmic Scales

Logarithms are used to increase the range over which numbers can be seen in a meaningful way. Logarithms are particularly useful when the data extends from the very small to the very large. There are many examples of there use:

Astronomy - the magnitudes of stars
In chemistry, the pH scale is a logarithmic scale which extends over 15 orders of magnetude, measuring the concentration of H ions in a solution. pH 0 is 10,000,000, De-ionised water is pH 7, 1 and pH 14 is 1/10,000,000
Acoustics - the intensity of sound is measure in decibels (dB). dB= 10 log₁₀(I/I₀) where I is the intensity of the sound measured in W m^-2, I₀ is defined as 1x10^-12 W m^-2.
Seismology - the Richter scale. An earthquake that measures 5 on the scale is ten times as powerful as a magnitude 4 earthquake, which is ten times as powerful as a magnitude 3 earthquake and so on.
Engineers often use a logarithmic scale (dBm) to measure the output power of photonic devices.
The Palermo Technical Impact Hazard Scale - a log scale that measures the probability of the Earth being struck by a meteorite of size. a Palermo Scale value of -2 indicates that the detected potential impact event is only 1% as likely as a random background event occurring in the intervening years, a value of zero indicates that the single event is just as threatening as the background hazard, and a value of +2 indicates an event that is 100 times more likely than a background impact by an object at least as large before the date of the potential impact in question.^[1]
Casualties of War^[2] A log scale to measure the total number of deaths in different conflicts. A single death would correspond to a magnitude 0. A terrorist campaign that kills 100 people would correspond to a magnitude of 2. The second world war had a total of 62,000,000 casualties, which would correspond to a magnitude 7.79. The ratio of the number of deaths is easy to compare, for example in the Viet-Nam conflict, the US lost about 50,000 soldiers, (magnitude 4.69), in the civil war of Independence, the US losses were 6824 (magnitude 3.83). The ratio of casualties is therefroe, 4.69-3.83 = 0.864 which is 10^0.864 ≅ 7.3 times.
The Lottery of Life
It has even been mooted that we should adopt a logarithmic risk scale. The disadvantage of this, however, is that relatively few people understand logarithms well enough not be confused. However, it has been suggested^[3], that people already understand the 'risk' of winning different prizes from the National Lottery. The probability of each greater prize roughly increases logarithmically. The probability of an event is matched with corresponding likelyhood of drawing so many balls, given a £5 stake. This could be particularly useful for medical practictioners, who have to convey a wide range of risks to the general public. The logarithmic risk is given by,
10 - log₁₀(1/P(x)) where P(x) is the probability of the event occuring. Table 1. sets out the probability of winning Lottery scales, probability, verbal risk scale and log risk scale.
Using this scale, a highly probable event is comparable to drawing 3 matching balls, or a 9 on the log scale. A moderately likely event compares to the chance of drawing 4 matching balls, or 8 on the log scale and so on. A highly unlikely event compares to drawing 6 balls, which carries a 'risk' of 1 in 2,796,763. (£5 stake remember) or less than 4 on the log scale. In one year, assuming you enter the 2 draws per week, this is 104 draws a year. Therefore, the chance of winning in one year on a £5 stake is 104/2796763 = 1 in 26892. or 5.57 on the log scale.
No. of Balls Probability P(x) Verbal scale Log Scale
3 1 in 11 High 9
4 1:206 Moderate 8
4+B 1 in 8878 Low 7
5 1 in 11 098 Very low 6
5+B 1 in 466,127 Minimal 4-5
6 1 in 2,796,763 Negligible <4
B = bonus ball
Table 1. The risk of winning the UK National Lottery.
Another unlikely event is being murdered. It is a difficult risk to assign. It depends on many factors, such as your age, social-class, etc. In the UK, a figure of 51 in 1,000,000 has been suggested as a the risk of being murdered in a year, which equates to a probability of 1 in 19,600. (5.7 log scale)
People place bias on good events over bad events (or bad events over good, if you work in the media) which distort their ideas of risk. I meet people that play the National Lottery and believe that they stand a good chance of winning the jackpot but the same people do not go out thinking, 'Ha-ha. This could be my lucky-day!' about being murdered so I have my doubts about the usefulness of this log scale.

No. of Balls	Probability P(x)	Verbal scale	Log Scale
3	1 in 11	High	9
4	1:206	Moderate	8
4+B	1 in 8878	Low	7
5	1 in 11 098	Very low	6
5+B	1 in 466,127	Minimal	4-5
6	1 in 2,796,763	Negligible	<4
B = bonus ball

References

[1] "Quantifying the risk posed by potential Earth impacts" by Chesley et al. (Icarus 159, 423-432 (2002)).

[2] Brian Hayes, “Statistics of deadly quarrels,”Am. Sci.90, 10–13 (Jan.–Feb. 2002).

[3] D M Campbell, "Risk language and dialects Expressing risk in relative rather than absolute terms is important", BMJ. 1998 April 18; 316(7139): 1242.