International projectionist (Jan-Dec 1939)

applies to the characteristic alone and not to the mantissa. We've gone forward, and we're now able to write the logarithms of any wholly decimal numbers : Character Mantissa istic (from (from Complete No. inspection) table) Log .9875 1 .99454 1.99454 .09875 2 .99454 2.99454 .009875 3 .99454 3.99454 .0009875 4 .99454 4.99454 Right now is a propitious moment to digress a whit, take a new test tube off the shelf, smear a slide and, peering through the 'scope, take a good look at a logarithm's close blood relation — antilogarithms. Every logarithm has an aunty. Up to this point we've started with a number, and from the tables and inspection found the complete logarithm of that number. Finding the antilogarithm is the reverse procedure. It's easy. Suppose you have the logarithm 2.81298. Take a look at the mantissa, go to the tables and find the sequence of numbers in the antilogarithm is: 6501. Now look at the characteristic, 2. The little minus-sign bonnet indicates the antilogarithm is wholly decimal, and the magnitude of the characteristic shows that there is one zero between the decimal point and the first digit which is not a zero. Hence the complete antilogarithm may be written: .06501. For the sake of good measure, let's tabulate several logarithms and their kinfolk. It might be a good hunch to place a card over the antilogarithms tabulated below and find them yourself from the table. If your antilogarithms agree with mine — well, your antilogarithms agree with mine. Logarithm Antilogarithm 1.65706 45.40 3.98318 .00962 4.20276 15950. 0.57066 3.721 So much for the essential nature of logarithms and antilogarithms, and how to find them. Now the thing to tackle is how to use the critters. Logarithms may be used for: 1. Obtaining the value of the product of numbers. 2. Obtaining the value of the quotient of numbers. 3. Obtaining the value of a number raised to any power. 4. Obtaining the value of any root of a number. The rule for finding the product of numbers is this: The logarithm of the product is equal to the sum of the logarithms of the individual numbers to be multiplied. Writing this rule symbolically we get: log [(NJCN.MNJ] = log Nj + log N2 + log N3 For a specific example, find the product (772) (.906) (2.45). log 772. = 2.88762 log .906 = 1.95713 log 2.45 = 0.38917 3.23392 Look up the antilogarithm of 3.23392 and you have the product. But on trying to find the mantissa .23392 in the table you'll run smack into a snag: you'll find mantissas close to it. There is .23376, corresponding to the sequence 17130; and there is .23401, corresponding to the sequence 17140. Thus we surmise, and correctly, that the sequence of our antilogarithm lies between 17130 and 17140. For the time being we are ignoring the location of the decimal point in our antilogarithm, considering the sequence of numbers in it alone. Organizing what we have, and calling the unknown and desired sequence S: Sequence Mantissa 17130 .23376 5 .23392 17140 .23401 The numerical difference between two sequences is proportional to the numerical difference between the' corresponding mantissas. Hence, from simple proportion we can write: -17130 .23392 —.23376 17140 —17130 .23401 —.23376 Solving this we get S = 171364. Spotting our decimal point from inspection of the characteristic, 3, our product is 1713.64. The foregoing operation employing proportion is known as interpolation. Before leaving interpolation we should also consider the case where the unknown is the mantissa instead of the sequence. For example, find the mantissa of 18254. The mantissa for the Pic Biz Ills No. 7241 Request from U. S. exhibitors that Paramount withdraw from radio programs such stars as Bing Crosby and Jack Benny, whose air appearances the exhibs charged hurt film business terrifically, was met by bland statement from Paramount that since the popularity of these stars was originally built up on the air the producing company was unable to do anything about it. sequence 18250 is obtainable directly from the table, as is also the mantissa for the sequence 18260; but the mantissa for the intermediate sequence 18254 is not given, and must be obtained by interpolation. Tabulating as before, but this time with M the unknown: Sequence 18250 18254 18260 Mantissa .26126 M .26150 Again, from simple proportion, we write : 18254 —18250 M .26126 18260 —18250 .26150 —.26126 Solving, we obtain M = .261356. So much for interpolation. Comes now division by logarithms. The rule is: The logarithm of the quotient is equal to the logarithm of the divisor subtracted from the logarithm of the dividend. In general terms: los N, N0 = log N,. — log'N, Let's start with an easy example. Find the quotient of 8472 divided by 25.01. log 8472 = 3.92799 log 25.01 = 1.39811 2.52988, from subtraction The quotient is the antilogarithm of 2.52988, which is interpolated to be 338.75. That was easy because the characteristics are both positive, and the mantissa of the dividend is larger than the mantissa of the divisor, enabling us to subtract the latter from the former without fuss or feathers. But the following problem in division is a little more intriguing. Find the quotient of .2501 divided by 84.72. log .2501 = 1.39811 log 84.72 = 1.92799 To handle this, add and subtract 10 to the characteristic of log .2501. This is permissible, for you can add 10 to something, then subtract 10, and you haven't changed the value of that something. Performing this operation gives: log .2501 = 9.39811 — 10 log 84.72 = 1.92799 7.47012 — 10, from subtraction .2501 Thus log = 7.47012 — 10 84.72 = 3.47012 The antilogarithm of 3.47012 is 18 INTERNATIONAL PROJECTIONIST