International projectionist (Jan-Dec 1939)

The Fundamentals of Mathematics By GEORGE LOGAN SOUND DEPARTMENT, METRO-GOLDWYN-MAYER STUDIOS Conclusion: Logarithms. It will be a help if the reader digest each article as it appears, for the ideas presented in subsequent sections hinge upon an understanding of topics discussed in earlier sections. Further, it is desirable that the issues of this series be cached away after reading, as back-reference may be~ useful before the series is completed. THERE is with me the vague recollection of reading, long ago, that logarithms were first exposed by some ancent Scottish doctor. I'm not going to check on this, so I'm probably inviting a barrage of contradictions. Nevertheless, that thought as to their origin sticks with me, and as a result I've always had the latent impression of a logarithm being a kind of bug, a sort of benevolent bacillus, if you please. The very word "logarithm" heightens this connotation; it has a biological sound to my ears. If imagination be allowed to run bersekedly rampant, one can just picture a grizzled old Scot medico, surrounded by pickled things in bottles, his gnarled knees showing under kilts, perched over a microscope and peering at something wriggling on a glass slide — lo! ... a logarithm. A logarithm's anatomy is made up of two and only two components a body, called the mantissa (not mantilla, that's something Spanish gals wear) ; and a head, called the characteristic. The mantissa is always positive. But the characteristic is wilfull, and may be positive or negative. A logarithm is a pretty straightforward bug when his characteristic is positive; but he's apt to be a little treacherous when he shows up with a negative characteristic. Never mind. We'll isolate him and show his workings in both forms. We'll take a good inoculatory shot of logarithms in the following discussions and wind up, we hope, invulnerable to any mystery they have exhibited in the past. Supplementing the needle, better dig out that cobwebby table of logarithms gathering dust on the shelf. A pencil and pad will be handy too. First of all, what is a logarithm? The definition is arbitrary, and resolves from giving names to the perfectly general algebraic expression: b1 = n Here x is defined as the logarithm of n to the base b. Stating it in another way, the logarithm of a number to a given base is the power to which the base must be raised to produce the number. From this it is apparent that logarithm tables could be produced for practically any base: say, b = 1, b = 2, b = 3, and on, and on. Actually, however, the work of creating such tables would be superfluous, for mathematicians have found that only two bases are necessary for the handling of problems: logarithms to the base 10 comprise one system; logarithms to the base 2.71828 comprise the other. Logarithms to the base 2.71828 are called natural logarithms, and are used principally in higher mathematics — calculus, hyperbolic functions, and sundry other headaches. Logarithms to the base 10 are called common logarithms, which soubriquet is well chosen, for they are the most commonly used for numerical calculations. For that reason we'll confine our microscopic study to the species log10 and postpone research on log2.7i82g to some future scribbing. With our jottings thus far you should be able to partially build up a common logarithm table yourself, assuming you were out in the woods, remote from one, and couldn't think of anything better to do in the woods. As a starter, what is the logarithm of 1? Stymied? Well, we know that: b* = n And that our base, b, is 10, and that our number, n, is 1. Hence: 10* = 1. The power (logarithm) to which 10 must be raised to produce 1 is zero. Hence x, the logarithm of 1 to the base 10, is zero, for: 10° = 1. Following the same procedure the logarithm of 10 is found to be 1.00000, for: 10 1.00000 — io Carry on in this wise and you'll find that the logarithm of 100 is 2.00000; the logarithm of 1000 is 3.00000; and the logarithm of 10,000 is 4.00000. Each of these complete logarithms is composed, as has been mentioned, of two parts: a characteristic and a mantissa. The portion of the logarithm to the left of the decimal point is the characteristic; the portion to the right is the mantissa. From this little trial shot at building a logarithm table certain deductions are naturally drawn, my dear Watson. These are: The characteristic is an integer of magnitude one less than the number of digits to the left of the decimal point in the number whose logarithm is sought. The mantissa depends only on the sequence of digits in the number, and does not depend in any way upon location of the decimal point in the number. Thus the mantissa is an independent sort of bug, and it doesn't give a tinker's expletive whether the number whose logarithm is sought is, for example, 7734 or 77.34, the mantissa is the same for each. Consequently, logarithm tables give simply the mantissas for particular sequences of digits. And the table leaves it up to you to prefix the mantissa with the proper characteristic, blandly assuming you know the rule for determination of the characteristic. Suppose we take the sequence 9875, progressively sprinkle it with decimal points, and find the complete logarithms for the numbers resulting: Character Mantissa istic {from {from Complete No. inspection) table) Log 9875 3 .99454 3.99454 987.5 2 .99454 2.99454 98.75 1 .99454 1.99454 9.875 0 .99454 0.99454 Now let's carry on with this decimalpoint-moving still further. The next step gives us the number .9875. Here we must pause to write another general rule: When the number whose logarithm is sought, is entirely a decimal, the characteristic of the logarithm is negative. Further, the magnitude of the characteristic is one plus the number of zeros between the decimal point and the first digit in the number which is not zero. Apphcation of this rule gives us — 1 as the characteristic of the logarithm of .9875. Negative characteristics are preferably written with the minus sign above, thus 1, for the minus sign OCTOBER 193 9 17