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6. If the mantissa be not found exactly in the table, take out that next less than the given one. Write this under the given one, at least as far as the figures are not common, and subtract it. With the difference thus found, enter the second column of the tablet marked "Pro." and the number adjacent to it is the sixth figure of N, and the first five those belonging to the next lower mantissa. The unit-place is, as before, to be assigned from the characteristic.

Thus, if log N

2-8730158, and we require N; take out the mantissa, and write it under the given one, according to the following type.


24 mant. of 74647

34 pp. 6 nearly, from the tablet ;

and we have 746476 for the number, so far as the mantissa is concerned; and from the characteristic we have the first figure in the second place to the right of the unit-place. Hence the number sought is '076476 very nearly.


1. Multiplication by logarithms. Add the logarithms of all the factors together, and the number whose logarithm is the sum will be the product. Theor. 1, p. 249.

Note 1. When some or all the characteristics are negative, the rules of algebraic addition must be employed.

Note 2. When the given numbers are of six places, it will not be necessary to do more to each of them in the way of correction separately, than to write the corrections down beneath the corresponding logarithms of five places, and at last add all the corrections and logarithms into one sum.

Ex. Find the product of 002356, 47·2985, ·32986, 42·7579, and 00004965. log 002356 = 3.3721753 p. 33, Tables.

log 47.2980

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}(to five places) p. 80.

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Hence, by the characteristic, the fifth decimal place is the significant figure, and the number composed of the digits 780348, the product itself is '0000780348 nearly.

This example contains instances of every possible variety of case that can

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4. Find the product 2876.9 × 10674 × 098762 × 0031598. Ans. '095830.

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2. Division by logarithms. Subtract the logarithm of the divisor from that of the dividend: the remainder is the logarithm of the quotient. See Theor. 2, p. 249. Note. In accordance with what is shown at p. 253, we may use the arithmetical complement of the subtractive logarithms, which will often much facilitate the operation. To effect this, instead of taking the several figures of the logarithm from the table, write (which can be, with little practice, done by inspection) the complement of each figure of the logarithm from 9 except the last, and the complement of this from 10.

For example, find the arithmetical complement of log. 37·5 and of ·00375.

log 37.5



log 00375 3.5740313

ac. log 37.5 = 8'4259687 ac. log 00375 = 12.4259687

in which, without writing down either of the lines, the arith. comp. may be written down from the inspection of the logarithms themselves in the table. Had there been six figures, the correction for the sixth might have been subtracted from the result of the addition, as in example 1. which follows.

This method of work should be early and regularly practised, on account of its almost constant occurrence in trigonometrical calculations.

When, however, there are several subtractive logarithms, it will be better for the most part to add them into one sum, and place the arithmetical complement of the whole under the column of additive ones, as in Ex. 2.

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4 nearly,

and characteristic 4 gives five places of integers: hence the quotient is 15621.4 nearly.

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3.1416 x 82 x 7. 02912 × 751.3 × DIT 3.1416 x 82 x 73 x 941

02912 × 751.3 × 6 × 41


82 1.9138139

73 1.8633229


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Or thus, and better, by the arithmetical complements : log 3.1416 = 0.4971509

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6. Divide 06314 x 7438 x 102367 by 007241 x 12.9476 x 496523, and compare the result with the product 8-71979 × 057447 × ⚫0206168.

Ans. They ought to be identical, or within a unit in the last place. 7. Divide 0067859 by 123459. Ans. 0000000549648. 3. Proportion by logarithms. This is only the application of logarithms to the operations of multiplication and division implied in finding the fourth term.

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Ex. 1. If 12.678: 14.065: 100·979 : x, then x = 112.027. 2. If 19864: 4678 50 4567: x, then = 11.8826. 3. If 498621: 2·9587:: 29587 x, then x = 17.5562.

4. Involution by logarithms. In conformity with what is shown in theor. 3, p. 249, we have log a" = n log a; which gives the process :

Multiply the logarithm of the base by the index of the power: the product is the log. of the power.

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5. Evolution by logarithms. Divide the log. of the number by the index of the root: the quotient is the log of the root: theor. 4, p. 249, where it is shown

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The only difficulty that can present itself is where the characteristic is negative, and not divisible by the index of the root. To remove this, add a negative number to the characteristic sufficient to render it the next higher multiple of the index, and add the same number taken positively to the positive part of the logarithm, that is to the mantissa. The quotient of the characteristic will in this case be a negative integer, and the quotient of the positive part of the expression will be decimal, and form the mantissa of the required logarithm. The following example will illustrate this.

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and the process obvious. In fact, involution and evolution by logarithms are the same rule, just as in common algebra under the same circumstances.


Expressions, 365-567a, 2-987631, 967845, 098674+ (273)3, and (1727)3.

21 373.

Values 19.1198, 1.44027, 9918624, 718315, 146895, and 1937115.


1. Find the values of 3.1416 x 82 x and 02912 × 751.3 X

73 41'



2. Find ✯ in 7241 : 3·58 :: 20-46: a, and in √724 :i

√5:6·927: x.

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of 00006 and show whether they be all real or not.

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001, and the third root

3172. 32.53


and of3× 034 √13.

9. Of how many figures does the number represented by 224 consist? And of how many does 99 consist?

10. Which is the greatest and which the least of the three numbers, 101o, 911, or 119? And show how to determine generally which is the greater, a' or b", supposing a greater than b.

11. Find the logarithm of 22·5, having given the logs of 2 and 3. 12. Having given the logs of 6 and 15 to find those of 8 and 9.

9 13. Given the logs of 2 and 3 to find those of and 16


Given the logs of 2, 3, 13, to find those of (24)



and 1/1.625.

15. Given log, 15 = 2·7080502, log, 5 = 1·6094379, to find log, 25. 16. Given log, 26931472, log, 5 = 1.6094379, and log10 1'9 = 2787536, to find log, 0019.


An exponential equation is one in which the unknown appears in the form of an exponent or index. When in this form, unmixed with other combinations, the solution is readily obtained by means of logarithms. Thus, if a* = N, then log N x log a log N, and x =

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Moreover, it makes no analytical difference how complexly the base or the index be given, the same method of solution applying to these as to the simpler form given above, provided the combinations be by multiplication, division, involution, or evolution only.

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