Logarithms

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Today I want to give a painless introduction to logarithms, a subject most people dislike so much they block it from their memories of high school algebra class.

What do music, earthquakes, social networks, and income inequality have in common? All involve multiple scales, i.e. numerical quantities that range from very small to very large. Logarithms provide a very convenient way of discussing these kinds of phenomena.

Pitch and loudness in musical notes, the destructive power of an earthquake, the number of friends a person has on Facebook, and the distribution of income by person, city, or country - all involve measurements that can vary by many orders of magnitude. The numbers involved could be small, like 1 or 2, or medium sized, like 1000, or very large, like 1,000,000 or more. It is awkward to try to compare such different numbers. In many disciplines, people have invented alternative "scales" to describe them: for instance, decibels measure sound intensity, while the Richter scale measures earthquakes. The intent of these scales is to convert the huge range of numbers down to something manageable by a person, such as the range from one to ten.

Let's start with musical sounds. The underlying physicalconcept is vibration: starting in a string or tube or drum,transmitted through the air, and finally making your eardrum vibrateas well.

Scientists can measure how powerful a sound is; this is relatedto how hard it pushes on your ear. If the sound is too powerful, itcan damage your hearing; if the sound is too weak, you may not be ableto hear it at all. Define the softest sound you can hear as 1unit of power. A whisper is 1000 on this scale.Normal conversation is 1,000,000 on this scale: much morepowerful than a whisper! Noisy traffic is 1,000,000,000. Soundsmuch louder than that (e.g. gun shots) can cause hearing damage.

Notice how large these numbers get. It is quite inconvenient to keeptrack of all the zeros! That's why people invented a simplerscale: decibels (dB). It works like this: count the number of zerosand multiply by ten.

For example, a whisper is 1000 times louder than the softest audiblesound, so we count 3 zeros, multiply by 10, and call it 30 dB.Conversation is 1,000,000 times louder, so we count 6 zeros and callit 60 dB. Loud traffic has 9 zeros, so it is 90 dB louder than thethreshold of hearing.The Decibel (Loudness) Comparison Chart has other examples, as well as OSHAlimits on what levels are safe.

A one decibel increase in power is too minor to be perceived. Threedecibels is about the limit of human perception. A five decibel changeis clearly noticeable. A ten decibel change corresponds to perceivingthe sound to be twice as loud. That means a 20 dBincrease makes the sound seem four times as loud, and 30 dB makes it seem8 times as loud.

Technically, when we count the number of zeros after the one, we aretaking the base ten logarithm. We can reverse the process: weoften write 10^n to mean n copies of 10 multiplied together. Forexample, 10^3 is 1000, because it equals ten times ten times ten(10*10*10). Similarly, 10^6 is a short-cut way to refer to one million, or 1,000,000. Notice that 10^3 (i.e. 1000) has 3 zeros,so its base ten logarithm is 3. Similarly, 10^6 (i.e. 1,000,000) has 6zeros, so its base ten logarithm is 6. So now you see the pattern: thebase ten logarithm of 10^n is just 'n'. The decibel scale justmultiplies 'n' by ten, giving 10*n, giving us a second digit of precisionwithout having to resort to using decimal fractions.

The Richter Scale for earthquakes works the same way, except itskips the multiplication by ten and just uses the base ten logarithmdirectly.

On the Richter scale, 3 represents a minor earthquake, the sortthat a person might notice, but that does no damage. A magnitude 4earthquake can shake indoor items and make rattling noises, but damageis still rare. A magnitude 5 earthquake can cause damage to poorlyconstructed structures nearby. A magnitude 6 earthquake can be quitedestructive to anything nearby. Even larger earthquakes can devastateincreasingly large geographic regions.

As you move from n to n+1 on the Richter scale, the amplitude (height)of the waves (motions in the rocks, recorded on a seismograph) get tentimes bigger, and the amount of energy released goes up by afactor of about 30. The actual amount of damage varies with other factors,such as distance from the center, depth below ground of the center,and the kind of soil, so Richter scale measurements are just a roughguide to the potential damage. Nonetheless, they provide a convenientway to describe the measurements, much more so than specifying theactual amplitude as 10^n.

We can play the same trick with any measurement that varies over alarge range. For instance, logarithmic scales are useful whendiscussing the number of friends a person has on a socialnetwork. There is no common word like "decibel" for this scale, butthe idea is the same. In typical social networks, or in the Internet(a network of computers), it is common to observe people (orcomputers) with very different numbers of friends: some may have justone or two friends; some may have ten or twenty, some a hundred, somea thousand or more. Here is a Wikipedia image of a social network constructedusing Facebook friends as the links: a line between two namesindicates people who are friends.

Source: http://en.wikipedia.org/wiki/File:Kencf0618FacebookNetwork.jpg

This is typical of diagrams of social networks: some highlyinterconnected people and groups, but also many others with just a fewconnections. The "six degrees of separation" game works because ofthis structure: pick any two people on the planet, and you will besurprised how often they can be connected by a chain of six "friends":even people who keep to themselves often know someone with a widercircle of friends; one of those may know a "hub" (a person with a verylarge number of connections), and before you know it, you have accessto a great many others through friends of friends of friends. The"Linked In" professional networking site uses the same principle toexpand your list of 100 contacts into hundreds of thousands of people"in your network".

Just as before, we can use logarithms. The base ten log of 10^n is 'n',so instead of saying someone has 1, 10, 100, or 1000 friends, we cancould refer to them as being at level 0, 1, 2, or 3 on a"connectedness" scale.

The same idea works for income. In the United States, many people havean annual income between $10,000 and $100,000 (10^4 and 10^5). But afew people have much larger incomes: one million dollars a year(10^6), or ten million (10^7) or one hundred million (10^8). Ratherthan writing these in words, or with lots of zeros, it is much moreconvenient to just write the exponents: 4, 5, 6, 7, 8. This isespecially handy if we want to make a graph: we can easily put thenumbers 4 through 8 on the horizontal axis of a graph, but if insteadwe wanted to put 10,000 through 100,000,000, we would have a problem:the vast majority of the US population (the part from ten thousand to one hundred thousand) wouldbe squashed down to a tiny portion of the left end of the axis,indistinguishable from zero. It would be like looking for your houseon a globe of the world: the scales are just too different.

We can play the same trick using powers other than ten. In Powers of Two Back in Time, we looked back one second, two seconds,four, eight, and so forth: these are powers of 2, written 2^n to meann copies of 2 multiplied together. The base two logarithm of2^n is just n. We saw that quite modest values of n convert toenormous chunks of time: 2^25 seconds is roughly one year, but 2^35seconds is roughly one thousand years, and 2^45 is roughlyone million years: back to prehistoric times. 2^55 seconds isa billion years, taking us back to the days of single celledorganisms, long before dinosaurs or mammals; and 2^59 seconds is 18billion years, which is larger than the age of the universe: we cannotgo back that far! Once again, rather than write really big numbers,it is easier to talk about the logarithms,since they are familiar sized: often one to ten, certainly less thanone hundred.

We end with a more elaborate example, which also uses base two logarithms:musical pitch. Scientists measure the frequency of sound vibrations incycles per second, also known as Hertz (Hz). For example, the lowest(longest and deepest sounding) string at the extreme left end of apiano vibrates 27.5 times per second(see Piano key frequencies at Wikipedia), while the highest (and shortest)string at the extreme right end vibrates 4186 times per second. Thatis a lot faster!

If we divide 27.5 into 4186, we get 152.2, which might suggest thatthere should be 152 keys on the keyboard, equally spaced at multiplesof 27.5. This turns out not to be the case.

There are actually 88 keys on a piano keyboard, and their frequenciesare not evenly spaced at all. For example, the second note from theleft is at 29.1 Hz, not even 2 cycles per second faster than the 27.5one. But the second to last note on the right is at 3951 Hz, or 235cycles per second slower than the 4186 one.

What is going on? It turns out that what the human ear perceives asequal intervals of musical pitch is related to what the mechanicalmeasurement of vibration would see as equal ratios offrequency.

For example, every time you go up one octave in pitch,you double the frequency. The lowest note on the piano iscalled A, and the subsequent white keys are called B, C, D, E, F, G,after which they repeat back to A. This second appearance of A isone octave higher than the first. If you play two notes anoctave apart on a conventional western musical instrument like thepiano, they blend very harmoniously. This is because a note withfrequency f also produces frequencies 2f, 3f, 4f, and so on; the notean octave higher produces 2f, 4f, 6f, and so on, which blends inperfectly.

William A. Sethares wrote a fascinating book, Tuning Timbre Spectrum Scale. He demonstrates unconventional (synthesized) instruments thatproduce sounds in non-integer ratios (e.g. f, 2.1f). They do not soundgood when you play notes an octave apart, but they do soundgood when played using wider intervals more appropriate to theirsound spectrum. But that's a topic for another day.

If we look at all the A keys on the piano keyboard, and write F=27.5for the frequency of the first one, then the frequencies of the othersare at 2*F, 4*F, 8*F, 16*F, and soforth. Here is a littletable that summarizes this:

n	0	1	2	3	4	5	6	7
2^n	1	2	4	8	16	32	64	128

The base two logarithm has a very useful property. Moving up by one,say from n to n+1, doubles the underlying frequency. This isexactly what we want for describing octaves.

Perhaps not surprisingly,we can calculate the frequencies for the other notes on the piano bythe same process. Each octave contains 12 notes. We have seen that infrequency terms, those notes are not equally spaced. But in pitch, orlogarithm terms, they are.

To fit 12 notes inside the octave from n to n+1, they must each be1/12th of the octave apart. To calculate the corresponding frequencies,we need to know how to deal with logarithms that are not wholenumbers.

Up to now, we said that you can find the base ten log of 10^n bycounting the number of zeros (i.e., n). And the base two log of 2^n isalso n. However, what do we mean by a logarithm that is not a whole number but afraction like 7/6 or 1/2?

It turns out that raising a number to the 1/2 power is the same thingas finding its square root. That's because of a very basicproperty of exponents: (x^a)*(x^b) = x^(a+b). When we multiply twonumbers, their logarithms add. This is clear for whole numbers,since x^a just means 'a' copies of x multiplied together; multiply by'b' more copies of x, and altogether you have 'a+b' copies.

Now consider the 1/2 power. x^(1/2) * x^(1/2) = x, since x^1 is just x(one copy, nothing to multiply it to). That means x^(1/2) is thenumber which, when multiplied by itself (i.e. when squared)gives back x. So x^(1/2) is the square root of x.

For example, 100 * 100 = 10,000. In power notation, 10^2 * 10^2 =10^4. This means that 100 is the square root of 10,000.

Suppose we need to find the square root of a number like 2 that is nota perfect square. We can try guessing: 1.5^2 = 2.25, a little too big,but 1.4^2 = 1.96, a little too small, so the square root of 2 isbetween 1.4 and 1.5; indeed, it is actually 1.414214 to six decimaldigits.

Ridgway Scott wrotea great bookon Numerical Analysis which starts out with a discussion of how yourcalculator can automate this process and calculate square rootsfor you rapidly. Using the techniques of numerical analysis, peoplehave developed ways to programcalculators and computers to find other roots and powers, and to findlogarithms of numbers that are not exact powers of ten (or powers oftwo).

For music, we just need one value: the 12th root of two. This ist = 2^(1/12) = 1.059463. It is a twelfth root in the sense that if we taketwelve copies of this number 't' and multiply them together, we findthat t^12 = 2.

We said a moment ago that to fit 12 notes inside the octave from n to n+1, they must each be1/12th of the octave apart. The 12 notes include both white and blackkeys. The white keys have letter names, and the black keys have morecomplicated names involving "sharps" and "flats", but we will ignorethem for now. The point is that when looking at a key, you have tocount both white and black notes to see how to number it.

The third key on the pianofrom the left is B, since it is the first white key after theinitial A. What is its pitch? Well, it is 2 notes past A, so inlogarithm terms, it will be at position 1+2/12 = 1+1/6 = 7/6. Whatfrequency does this correspond to? A=1 was 27.5 Hz; the A above that,at 2 on our log scale, doubles that frequency to 55 Hz. For keysin-between, adding 1/12 on the log-two scalecorresponds to multiplying by t = 2^(1/12).So, starting from 1 and adding 2/12 in "pitch space" corresponds tostarting at 27.5 Hz and multiplying by 1.059463 twice, in"frequency space". That gives 30.87 Hz, which is indeed the frequencyfor the lowest B on the piano.

You can calculate the frequency of any note onthe piano this way: they are equally spaced in pitch, i.e. in the basetwo log of frequency; with twelve per octave, if you move up by 'k'keys, you multiply the frequency by t^k, where t = 2^(1/12).

That's enough for today.Music is a fascinating subject, with many more connections tomathematics, but we will have to wait for another time to explore them.Hopefully, you now have a more intuitive sense for what logarithmsare and why they are useful in so many different contexts!

If you liked this article, you may also like Modular Arithmetic, Prime Numbers, and a Little Bit of Cryptography, or perhaps Counting and Number Systems.

Or, if you know calculus and want to learn how calculators actually compute logarithms numerically, you may like Numerical Analysis: How to Calculate Special Functions.

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