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(10) A man standing 500 feet from a building observes a flag pole on the top. The angle of elevation of the bottom of the pole is 27°
and that of the top is 29.5°. Find the height of the pole and of the building
28.2 feet and 254.75 feet. Answers. (11) Find ZA, correct to nearestio and find AC in figure 124.
(12) The estimated trajectory of a bomb dropped from 20,000 feet will intersect the target 2 miles away. What will be the angle of
depression from the plane to the target at the time the bomb is dropped?
62.2° Answer. (13) On a certain observation maneuver a squadron of 18 airplanes, figure 125, was ordered to search an area of ocean starting from a base on the coast and fanning out radially from the base. Visibility for this maneuver was estimated to be 20 miles. A 5-mile overlap was necessary for optimum efficiency. Assuming a straight coast line, at what distance from their base would the operation begin to be ineffective?
72 Introduction to logarithms.
75 Use of tables...
77 Negative characteristics
78 Multiplication and division.
79 Powers of numbers.Extraction of roots..
81 Application to right triangles.
82 Evaluating formulas with logarithms-
83 Oblique triangles
84 72. Purpose. The use of logarithms presents a convenient method for performing arithmetical calculations with ease and rapidity. It will be seen that logarithms are exponents, or powers, of ten, and hence that the rules for the combination of exponents hold for them. Therefore calculations involving the operations of multiplication, division, raising to powers, and extraction of roots can be performed with a sufficient degree of accuracy and with little effort.
73. Introduction to logarithms.-a. Find the values corresponding to the positive integral powers of 10:
104=10,000 etc. b. This tabulation suggests the question: For any number greater than one, say 50, is there a number which may be used as an exponent so that 10 to that power will equal 50? From the table, since 50 is between 10 and 100, it seems reasonable to suppose that this exponent is between 1 and 2 (or 1 plus a decimal if it exists). At a later stage in mathematics, it is proved that such a number does exist and that its value is between 1 and 2. Again, since 576.3 is between 100 and 1,000, note that the exponent of 10 should be between 2 and 3; that is, 102.+=576.3. These exponents of 10 are called logarithms.
74. Logarithms.—The logarithm of any number to the base 10 is that exponent which causes 10 to that power to be equal to the given number. Thus, since 103=1,000, the definition states that log 1,000= 3. These two equations are equivalent; they are merely two different ways of stating the same fact (the relation between 3, 10, and 1,000). It happens, however, that the logarithmic statement is more convenient for computational purposes. Again, since 101.+=50, it can be said that log 50=1.+ (some decimal fraction).
a. Logarithms (and all numbers for that matter) may be considered to be composed of an integral (or whole numbered) part and a decimal fraction part. The integral part of the logarithm is called the characteristic and the decimal part, the mantissa. Consider now the numbers 600, 105.3, 732, 986.2; referring to the tabulation in paragraph 72, each of these four will be 2 plus a decimal part, that is, their logs all have 2 as their characteristic, as also is the case for all numbers between 100 and 1,000.
b. This indicates the following rule which MUST BE MEMORIZED: Any number greater than one has a logarithm whose characteristic is one less than the number of digits to the left of the decimal point.
c. The consideration of logarithms of numbers less than one will be deferred until later.
d. Exercise.—(1) What are the characteristics of the logs of the following numbers: 62; 385; 78.34; 67.823; 4; 5.78; 10,000; 7,683; 17?
(2) State between which powers, of 10 the numbers whose logs have the given characteristics lie: 0; 1; 2; 3; 4; 5; 6; 7; 8.
e. When a particular number is given, say 27.38, there are given two things; a certain set or sequence of digits, and the location of the decimal point. It has been seen how intimately the characteristic of a log of a number is associated with the position of the decimal point in the number. One might therefore expect (and it is true) that the mantissa or decimal part of a logarithm is dependent upon, or determined by, the sequence of digits in the number. In other words, two numbers such as 27.38 and 273.8 will have logarithms whose mantissas are the same; their logs will differ only in the characteristics.
75. Mantissa.-Mantissas in general are found to be unending decimals similar to the decimal equivalents of V2 and 1. The accuracy desired for a computation will determine the number of places to be used in the mantissa table; that is, if results to only four significant figures are desired, mantissas to four decimals are used, etc. The tables of mantissas have been computed by processes developed in more advanced mathematics; and, because of the mantissa and characteristic properties already mentioned, the table needs to be computed for only a certain range of numbers. It is in this fact that the number 10 finds its justification and practical value as the base for a system of logarithms.
76. Use of tables.-A four-place logarithms table is given in appendix II. Make it a definite policy, when finding the logarithm of a number, to write the characteristic first. Only after this is done, begin looking for the mantissa in the table.
(1) Example: Find the logarithm of 30. Write log 30=1.- as determined by the characteristic rule. The sequence of digits is 3,0,0,0. The first two digits 3,0, are found in the column headed at the top by “N.” Follow down this column to the number by 30. It is known that the third digit, the one after the decimal point, is 0. Looking in the vertical column headed “O”, the number 47712 is found opposite 30. This number is the mantissa of the logarithm of 30, that is, log 30=1.4771. This means according to the discussion that, to four significant places, 101.4771=30.00.
(2) Example: Find log 7,640. First, log 7640=3.—. Then find 76 in the column headed "N." In the row corresponding to 76, go over to the column headed 4; the number appearing there is 8831, the desired mantissa. Hence log 7,640=3.8831.
77. Interpolation.-a. If the fifth digit of a number is not zero, interpolation must be used to find the logarithm's mantissa.
Example: Find the logarithm' of 76,430. Since 76,430 is between 76,400 and 76,500, the mantissa of its logarithm will be between the mantissas of the logs of 76,400 and 76,500: in fact, one could estimate
3 it to be of the difference between the mantissas given in the table
10 beyond the smaller one appearing there. Previously it was found that log 76,400=4.8831; the difference in the table or (tabular differ
10 and rounding off to 2, one finds the mantissa of log 76,430 to be .8833. Then log 76,430=3.8833. This relation may be expressed as a formula, thus:
Last significant digit x tabular difference
10 b. If given a logarithm and the number which has it as its logarithm are to be found, apply the previous process in reverse order.
(1) Example: log N=1.9206, find N.
Since the characteristic determines the decimal point, leave it to the last. First find the sequence of digits corresponding to a mantissa of 9206. Hunting in the interior of the tables, one finds this to be 833. Then the rule for characteristics states that there must be two digits to the left of the decimal point. Hence N=83.3.
(2) Example: Log W=2.5748, find W. One finds that the mantissa .5748 is between .5740 and .5752. Corresponding to the smaller of these, there is the sequence of digits 375. In this case the adding factor and difference are known; they are 8 and 12. Then using the equation in a above, one gets,
8X10 Last digit= =6%=7, when rounded off. Hence W=375.7, 12
, the decimal point being determined by the rule.
c. Exercises.-Find the following logarithms:
1.6702 Answer. (3) log 68.5 (4) log 273.5
2.4370 Answer. (5) log 9 (6) log 81.32
1.9102 Answer. (7) log 7.008 (8) log 760.1
2.8809 Answer. (9) log 5010 Find the number which has the given log. (1) log B=1.7810 (2) log A=2.9576
A=907.0 Answer. (3) log E=1.8790 (4) log C=4.3521
C=22.50 Answer. (5) log F=3.3333 (6) log D=2.6969
D=497.6 Answer. (7) log H=.5724 (8) log G=.6008
G=3.988 Answer. (9) log K=7.1660
78. Negative characteristics.-a: Take for granted the definition (which can be justified) that a negative exponent means the reciprocal of the number affected by the same numerical but positive exponent. By this definition:
102 100 6. Now construct a table of negative powers of 10: