Reflections on Senior Mathematics

Shakes, quakes, and stars - Investigations with exponential and logarithmic functions

Mary Coupland and Jules Harnett, University of Technology, Sydney

Indices, logarithms, and exponential functions are widely used in scientific applications of mathematics as they provide models for a variety of physical situations. We think this can provide motivating examples in the teaching of these topics. In the workshop at the conference, we first investigated some activities based on everyday situations that could be used as starting points. We also provided some information that may go beyond the current school syllabus in mathematics, but which provides interesting background knowledge for teachers.

Investigations that use indices and exponential functions

I. Could you be related to a famous historical figure?

Considering only biological parentage, how many parents, grandparents, great-grandparents, great-great-grandparents & do you have? Could you draw up a table? What is the relation between n the number of  - grandparents you have, and g the number of generations back they are? How many years ago do you think your  - grandparents were alive? Find out where they were probably living and the population of that part of the world at that time. You may be related to someone famous! You may even be related to the person sitting next to you!

II   The disappearing function

Take a packet of M&Ms. Pour into a cup, shake, and tip out onto a plate, spreading them into one layer. Count how many have the M showing, then make these disappear, and count the number remaining. Keep track of these numbers and repeat until you run out of M&Ms. If x is the number of shakes, and y is the number of M&Ms remaining after each shake, plot your experimental values for x and y. Invent an equation that models this situation and draw it on the graph paper to see how closely it fits the experimental data.

Would more data give a graph that more closely fits the model? You could repeat the experiment (but all the M&Ms have disappeared!) or combine the class results somehow & what happens?

(Adapted from Stevens 1993)

III Bouncing back

How high does a super-ball bounce when dropped from 1 metre? See if you can verify the claim that they bounce to 80% of the height from which they are dropped. After x bounces from a height of 1 metre, what equation predicts their height y? What does the graph of this function look like? What is wrong with the illustration below?

Figure 1

IV  An old treasure mystery

In the Valley of the Kings in Egypt there are no soot marks from candles or lamps in some of the innermost chambers and yet there are intricate paintings on the walls. How did they get the light in there? It has been suggested that a series of mirrors could have been used. If each of nine mirrors reflects 40% of the light falling on it, how much light would there have been in the chamber? How could you check out this theory? Do you have an alternative theory?

(III and IV are adapted from Understanding Indices, Schools Council 1975)

    Figure 2

V    How large can a steak be?

Our photocopier has only one enlargement setting, so to get larger pictures we have to enlarge the enlargements. What is the enlargement ratio? How many times would you have to copy the photo to make it at least twice / three times / four times / five times as large as the original?

The transition to logarithms

The examples above all lead to equations of the form  with m an integer. In some cases, finding m for a given a and b is done by trial and error on the calculator. This leads to the question of how to find m accurately when b is not an integer power of a. Moving to logs can be done in several ways. For example, from the statement:

        'm is the index when b is written as a power of a',

go to    'm is the logarithm of b in base a',

with logarithm as just a funny word for the time being, and then to

              , as the notation used for changing the subject of the formula

Alternatively, an investigation of powers of ten including non-integer powers could be used to lead to the discovery of the log button on the calculator as being the bottom that 'undoes' the 'raising to a power of ten' button. This investigation could arise from plotting data that have a very wide range of values, for example, life expectancies of various living things, populations of different countries, and distances from the Earth of various planets, stars and galaxies. A third treatment could be the historical approach, noting that Napier, the inventor of logarithms, did not use index notation at all. A good source here is Great Moments in Mathematics by Howard Eves (1983), available from AAMT. A fourth possibility, useful for calculus students, is to investigate the area under the curve  from  to

Once students are familiar with logs and log laws, the question of who uses such things naturally arises, and presents a good opportunity to talk about logarithmic scales.

General introduction to the idea of a logarithmic scale

Human senses such as hearing and sight respond to extremely wide ranges of stimuli. When we try to represent these huge ranges in everyday terms, some conversion is necessary to make comparisons possible. Since everyday measurements tend to be made with linear scales, it is not surprising that methods for representing very large multiplicative scales by smaller additive scales are needed. In the history of western science, this need arose in the eighteenth and nineteenth centuries as instruments were developed to measure the intensity of various physical phenomena. Fortunately Napier (1550-1617) had already invented a scale which converted a geometric progression into an arithmetic progression. This enabled him to replace multiplication by addition, an invention that 'doubled the life of astronomers' (Eves 1983, p. 185) by reducing the time spent on tedious calculations. He called his invention logarithms, meaning ratio numbers. These numbers were just what the scientists needed. In fields as disparate as earthquake research and astronomy, logarithms provided the key to the representation of data that were spread over very large ranges.

Imagine a stimulus of strength  that is just detectable by human senses. Allocate this the value of 0 on our new scale. Now if another stimulus is created by K times the energy of the first, or reference stimulus, allocate it the value of 1 on our new scale. You could also think of it as being K times as intense as the reference stimulus. If a third stimulus is created by  times the energy of the reference stimulus, allocate it the value of 2 on our new scale, and so on.

This leads to the following table:

Strength of stimulus					...
New scale, n	0	1	2	3	...

The table could also be written as:

Ratio	1	K			...
New scale, n	0	1	2	3	...

We then see the formula that describes the relationship:

Finally, converting to base 10 logarithms gives:

                                                         ...(1)

where

       n    is the number on the new scale,

      is the strength of the reference stimulus,

       S   is the strength of the measured stimulus.

We will now see how formula (1) is used in three different physical situations.

Sound

The range of human hearing is impressive. If the intensity of sound is measured in watts per square metre, you can just hear a noise of  that is, 0.000 000 000 001 . You would suffer permanent ear damage if subjected to a noise of 100 . A logarithmic scale helps to make sense of this wide range of numbers. If K is chosen to be 10, then a tenfold increase in the intensity of the sound increases the perceived loudness by one bel, a unit named in honour of Alexander Graham Bell, in which case our original formula (1) would look like:

where

       n    is the number on the loudness scale in bels,

         is the strength of the reference sound: the faintest audible sound,

       S    is the strength of the measured sound.

It is thought that the smallest change in sound that a human can detect is one tenth of a bel, that is, a decibel. This gives us the unit that is actually used. One bel would equal 10 decibels. The relevant formula is then:

In this case D is measured in decibels. You can see that the result is the same as assigning the value of 0 decibels to the barely audible sound of intensity  and then each time the intensity of the sound is increased by multiplying by a factor of ten, the loudness of the sound as measured in decibel units is increased by adding ten decibels.

Here is a chart of some familiar sounds and noises: can you fill in the gaps?

Sound	Decibels	Intensity in
Threshold of hearing	0
A very very faint whisper (good gossip wasted)	10
Leaves rustling in a breeze	20
A soft whisper at 5 metres	30
Ordinary conversation	60
A loud singer about 1 metre away
Heavy traffic (inside the car)	85
A noisy kitchen with several appliances operating	100
A jet aircraft take-off at 600 m
A lawn mower	107
Inside a disco, band with amplifiers	117
Jet aircraft take-off at 60 m	120	1
Pneumatic riveter (also the painful level for humans)	130	10
Jet aircraft take-off at 30 m		100

Decibels are also used to express the input/output ratios of various electronic devices, in which case S may be a voltage or a current. They are also used to refer to the relative strengths of two signals.

Stars

In astronomy there are many applications of logarithms, for example classification of stars by their brightness, and the use of observational data to test cosmological theories. Exponential functions are used to model the thermal and non-thermal radiation from galaxies.

The story of classification of stars according to their brightness begins with the Chinese astronomer Shih-Shen, who catalogued 809 stars in the fourth century BC. His better known successor, the ancient Greek astronomer Hipparchus of Rhodes, invented a magnitude system of stellar brightness in the second century BC. Hipparchus proposed six brightness classes, class one containing the brightest stars, and class six containing the faintest ones visible to the human eye. (Karttunen et al. 1994, p.93)

By the 1850s, measurements of stellar brightness were possible and it was discovered that the difference between the first and sixth magnitude stars corresponded to a brightness ratio of roughly 100. (Snow 1991). This enabled Norman R. Pogson to propose a more accurate classification which is still in use today. Pogson refined Hipparchus' qualitative scale to give a quantitative measure of brightness. Since there were six classes and therefore five intervals, Pogson realized that the brightness ratio of one class to the previous one is given by  Today astronomers use the formula:

where

       m is called the stellar magnitude,

       F is the flux density, measured by photometry in  these days.

Note that the constant (2.5) is not just the rounding of  In fact it arises from our original equation (1) with K equalling  The negative sign accounts for the fact that as the flux density F increases, m decreases. This is all due to Hipparchus, who as you will recall, assigned the brightest stars to class one and the faintest to class six! Astronomy abounds in such arcane facts.

Listed below are some familiar astronomical objects with their magnitudes.

The stellar magnitude scale differs from the decibel system in several ways. As mentioned, it seems to us to be 'backwards', with brighter objects scoring lower on the scale. Further, the threshold of stimulus just detectable by humans is not set at zero but at 6, since that is the class of just-visible stars as decided by Hipparchus. The scale can be continued in both directions, as could the decibel scale; in fact the Hubble space telescope can detect stars as faint as magnitude 28.

Earthquakes

The famous Australian mathematician K.E. Bullen (1963) gave a brief outline of the history of measurement of earthquakes in the first chapter of his book An Introduction to the Theory of Seismology. Aristotle classified earthquakes into six types according to the different ways the ground shook; and in AD 132 the Chinese philosopher Chang Heng 'devised an artistic instrument for indicating the direction of the first main impulse due to an earthquake'. (p.1)

During the late eighteenth and early nineteenth centuries most earthquake studies were concerned with the geological effects and the effects on buildings. The Rossi-Forrel intensity scale was produced in 1878 and used to estimate surface effects of earthquakes as well as to determine isoseismal lines, which in turn were used to plot epicentres. Another qualitative scale, derived from descriptions of the nature of the disturbance and damage to objects, is the Mercalli scale which is still in use in the form modified by Wood and Neumann in 1931. (See Appendix 1.)

With the invention of the seismograph towards the end of the nineteenth century, more accurate measurements of disturbances in the Earth were possible. The development of wave theory in mathematics (Cauchy, Poisson, Stokes, Green, Kirchoff, Kelvin and Rayleigh, to mention a few names) contributed to the work of scientists who constructed theories about the structure of the Earth. The shock waves produced by earthquakes, and later on, large explosions, could now be recorded and measured and this empirical evidence was used to test theoretical models of the structures of the surface and of the interior of the planet.

One of the first attempts to develop a quantitative scale for earthquakes was made by Richter in 1935. He defined the magnitude of an earthquake as the logarithm (to base ten) of the maximum amplitude traced on a seismogram by a seismograph of certain specifications. There was some arbitrariness in the definition of magnitude but in the 1950s Richter's scale was refined and gave a value of 8.9 to the largest known earthquakes. Developments in the theory of seismology, largely due to Jeffreys, had made it possible to estimate the actual energy released by an earthquake, and Gutenberg and Richter produced the formula:

In this formula,

M is the Richter scale magnitude of an earthquake. (Bullen 1963, p. 271)

The Richter scale formula shown above is equivalent to the formula (1) we have used above, with K = 31.62 approximately. The general rule of thumb used by geologists is that a thirty-fold increase in the energy or intensity of an earthquake is represented by an additive increase of one in the Richter scale.

Another way to define the Richter scale is to use the formula:

which shows the Richter scale M for an earthquake of E kilowatt-hours (Shenk 1988). We leave it as an exercise for the reader to show that the geologist's rule of thumb can be derived from this formula. It is also interesting to compare the energy in an earthquake (for example, the Newcastle earthquake on 28 December, 1989 measured 5.6 on the Richter scale) with the electrical energy supplied by your local electricity provider. For example, the Sydney County Council in the St George and Sutherland area averages 2.5 gigawatts, so in 24 hours, say, the energy distributed would be  kilowatt-hours.

You might like to verify and/or complete the following calculations:

Earthquake	Magnitude (Richter scale)	Energy in kilowatt-hours
1906  San Francisco, USA	8.25	3.68 x
1964  Alaska	7.5	2.8 x
1971  Papua New Guinea	8.1
1976  Tangshan, China	7.6
1989  San Francisco Bay Area	7.1	7.07 x
1989  Newcastle, Australia		4.08 x
1995  Kobe, Japan		3.56 x

Appendix 2 contains information about earthquakes in Australia.

Other applications

Another logarithmic scale is the pH scale used in chemistry. If  is the concentration of hydrogen ions measured in moles per litre (1 mole =  molecules), for a particular solution, then the pH of that solution is defined to be  Distilled water is said to be neutral and has a pH of 7. Solutions with  greater than that of water are said to be acidic; those with  less than that of water are said to be alkaline or basic.

Besides logarithmic scales, various functions based on the log and exponential functions are widely used in science. Some of these are studied in 2 and 3 Unit mathematics and need to be well understood but others can be used to provide interesting equation-solving practice, even if time does not permit a more thorough study of the function and why it is used. The equations of the logarithmic scales described in this article could be used in that manner, although we enjoyed puzzling through the various forms, finding their common features, and believe that some interesting projects could be designed around them. The link with arithmetic and geometric progressions is fascinating and the frequent use of scientific notation and various units of measurement is something that science students always seem to need to practise.

References and further reading

Bullen, K.E. (1963). An Introduction to the Theory of Seismology, (3rd Edition), Cambridge University Press.

Eves, H. (1983). Great Moments in Mathematics (Before 1650), The Mathematical Association of America.

Jacobs, H.R. (1970). Mathematics, a Human Endeavor, W.H. Freeman and Co., San Francisco.

Karttunen, H. et al. (Eds) (1994). Fundamental Astronomy, Springer-Verlag, Berlin.

Schools Council (1975). Understanding Indices, Schools Council, Heinemann, UK.

Shenk, A. (1988). Calculus and Analytic Geometry, (4th Edition), Scott, Foresman and Company, Glenview, Illinois.

Stevens, J. (1993). Generating and analyzing data. Mathematics Teacher, Volume 86, Number 6, pp. 475-487.

Figure 4. Are you sure there isn't a simpler means of writing 'The Pharaoh had 10 000 soldiers'?
- Perhaps they could try logs!

Appendix 1

An abridged form of the modified Mercalli scale, with the corresponding Rossi-Forel scale numbers indicated in brackets. (From Bullen 1963, pp. 317-318)

 

Appendix 2

Information about Australian earthquakes from the leaflet Earthquake Awareness for Australians, produced by the Natural Disasters Organization.

Richter Scale: Note that energy of a magnitude 6.0 earthquake is about 30 times that of a magnitude 5.0, which is 30 times that of a magnitude 4.0, and so on.