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log. 10.4123 very nearly answer
1.017546 I multiply by 3 to find the 3d power, and divide by 5 to obtain the 5th root.
LI. There is an expedient generally adopted to avoid the negative characteristics in the logarithms of decimals. I shall explain it and leave the learner to use the method he likes the best.
1. Multiply 253 by .37. log. .37
1.568202 log. 253
log. 93.61 nearly answer
1.971323 Instead of using the logarithm 1.568202 in its present form, add 10 to its characteristic and it becomes 9.568202.
log. .37 log. 253
log. 93.61 as above.
1.971323 In this case 10 was added to one of the numbers and afterwards subtracted from the result ; of course the answer must be the same.
2. Multiply .023 by .976.
Take out the logarithms of these numbers and add 10 to each characteristic. log. .023
8.361728 log. .976
log..0224473 nearly ans.
2.351178 We may observe that, in this way, when the first left hand figure is tenths, the characteristic, instead of being I is 9, and when the first figure is hundredths, the characteristic is 8, &c.
. That is, the place of the first figure of the number reckoned from the decimal point corresponds to what the characteristic falls short of 10. Whenever in adding, the characteristic exceeds 10, the ten or tens may be omitted and the unit figure only retained.
In the first example, one number only was a fraction, viz. .37. In adding, the characteristic became 11, and omitting the 10 it became 1, which shows that the product is a number exceeding 10.
In the second example both numbers were fractions, of course each characteristic was 10 too large. In adding, the characteristic became 18. Now instead of subtracting both tens or 20, it is sufficient to subtract one of them, and the characteristic 8, which is 2 less than 10, shows as well as 7 would do, that the product is a fraction, and that its first figure must be in the second place of fractions or hundredth's place.
If three fractions were to be multiplied together, there would be three tens too much used, and the characteristic would be between 20 and 30; but rejecting two of the tens, or 20, the remaining figure would show the product to be a fraction, and show the place of its first figure. 3. What is the 3d
the 3d power of .378 ! log. .378
9.577492 Multiply by
log. .05401 nearly ans.
Multiplying by 3 is the same as adding the number twice to itself. The characteristic becomes 28, but omitting two of the tens or 20, it becomes 8, which shows it to be the logarithm of a fraction whose first place is hundredths.
If it is required to find the 3d root of a fraction, it is easy to see, that having taken out the logarithm of the fraction, it will be necessary to add two tens to the characteristic, for it is then considered the third power of some other fraction, and in raising the fraction to that power, two tens would be subtracted.
In the last example the logarithm of the power is 8.732476, but in order to take its 3d root, it will be necessary to add the two tens which were omitted.
For the second root one ten must be previously added, and for the fourth root, three tens, &c.
In dividing a whole number by a fraction, if 10 be added to the characteristic of the dividend, it cancels the 10 supposed to be added to the divisor. If both are fractions the ten in the one cancels it in the other; and if the dividend only is a fraction, the answer will of course be a less fraction. Consequently in division the results will require no alteration.
6. Divide 57 by .018.
log. 3166.7 nearly ans.
Here in subtracting I suppose 10 to be added to the first characteristic, and say 8 from 11, &c.
7. Divide .2172 by .006.
log. 36.2 Ans.
1.558709 In taking the arithmetical complement, the logarithm of the number may be subtracted immediately from 10. The logarithm of 2 being .301030, its arithmetical complement is 1.698970. Adding 10 it becomes 9.698970. It would be the same if subtracted immediately from 10 thus 10 - .301030 = 9.698970.
8. It is required to find the value of x in the following expression :
log. x = 1.13835 nearly
0.056262 Find the value of x in the following equations.
38% = 583. Observe that the 2d power of 38 is found by multiplying the logarithm of 38 by 2, the 3d power by multiplying it by 3, &c. which will give the logarithm of the result. Hence we have the following equation ; the logarithm of 38 being 1.579784 and that of 583 being 2.765669.
X 1.579784 = 2.765669
= 1.75066 +
1.579784 The value of x is found by dividing one logarithm by the other in the same manner as other numbers. It might be done by logarithms if the tables were sufficiently extensive to take
out the numbers. By a table with six places an answer cor· rect to four decimal places may be obtained.
In taking out the logarithms the right hand figure may be omitted without affecting the result in the first four decimals. log. 2.76567
0.441794 log. 1.57978
log. = 1.75064 +
LII. Questions relating to Compound Interest. It is required to find what any given principal p will amount to in a number n of years, at a given rate per cent. r, at compound interest.
Suppose first, that the principal is $1, or £1, or one unit of money of any kind.