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5. The 7th power of 0.9061 -is 0.5015. 6. The 5th power of 0.9344 is 0.7123.


47. Evolution is the opposite of involution. Therefore, as quantities are involved, by the multiplication of logarithms, roots are extracted by the division of logarithms; that is,


The reason of the rule is evident also, from the fact, that logarithms are the exponents of powers and roots, and evolution is performed, by dividing the exponent, by the number expressing the root required. (Alg. 257.)

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In the first of these examples, the logarithm of the given number is divided by 2 ; in the other, by 3.

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The division is performed here, as in other cases of decimals, by removing the decimal point to the left.

5. What is the ten thousandth root of 49680000 ?

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will occur,

We have, here, an example of the great rapidity with which arithmetical operations are performed by logarithms.

48. If the index of the logarithm is negative, and is not divisible by the given divisor, without a remainder, a difficulty

unless the index be altered. Suppose the cube root of 0.0000892 is required. The logarithm of this is 5.95036. If we divide the index by 3, the quotient will be – 1, with - 2 remainder. This remainder, if it were positive, might, as in other cases of division, be prefixed to the next figure. But the remainder is negative, while the decimal part of the logarithm is positive ; so that, when the former is prefixed to the latter, it will make neither +2.9 nor-2.9, but —2+.9. This embarrassing intermixture of positives and negatives may be avoided, by adding to the index another negative number, to make it exactly divisible by the divisor. Thus, is to the index – 5 there be added – 1, the sum – 6 will be divisible by 3. But this addition of a negative number must be compensated, by the addition of an equal positive number, which may be prefixed to the decimal part of the logarithm. The division may then be continued, without difficulty, through the whole.

Thus, if the logarithm 5.95036 be altered to 6 + 1.95036 it may be divided by 3, and the quotient will be 2.65012. We have then this rule,

49. Add to the index, if necessary, such a negative number as will make it exactly divisible by the divisor, and prefix an equal positive number to the decimal part of the logarithm.

1. Required the 5th root of 0.009642.
Power 0.009642 log. 3.98417

or 5+2.98417
Root 0.3952


2. Required the 7th root of 0.0004935.
Power 0.0004935 log. 4.69329

or 7)7+3.69329
Root 0.337


50. If, for the sake of performing the division conveniently, the negative index be rendered positive, it will be expedient to borrow as many tens, as there are units in the number denoting the root.

What is the fourth root of 0.03698 ?

Power 0.03698 4)2.56797 or 4)38.56797
Root 0.4385



Here the index, by borrowing, is made 40 too great, that is, +38 instead of -2. When, therefore, it is divided by 4, it is still 10 too great, +9 instead of – 1.

What is the 5th root of 0.008926 ?

Power 0.003926 5)3.95066 or 5)47.95066
Root 0.38916



51. A power of a root may be found by first multiplying the logarithm of the given quantity into the index of the power, (Art. 45.) and then dividing the product by the number expressing the root. (Art. 47.)

1. What is the value of (53)*, that is, the 6th power of the 7th root of 53?

Given number 53 log. 1.72428
Multiplying by


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2. What is the 8th power of the 9th root of 654?


52. In a proportion, when three terms are given, the fourth is found, in common arithmetic, by multiplying together the second and third, and dividing by the first. But when logarithms are used, addition takes the place of multiplication, and subtraction, of division.

To find then, by logarithms, the fourth term in a proportion, ADD THE LOGARITHMS OF THE SECOND AND THIRD TERMS, AND from the sum SUBTRACT THE LOGARITHM OF

The remainder will be the logarithm of

THE FIRST TERM. the term required.

Ex. 1. Find a fourth proportional to 7964, 378, and 27960.

Numbers. Logarithms.
Second term 378

Third term 27960


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2. Find a 4th proportional to 768, 381, and 9780. Second term 381

2.58092 Third term 9780


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53. When one number is to be subtracted from another, it is often convenient, first to subtract it from 10, then to add the difference to the other number, and afterwards to reject the 10.

Thus, instead of a-b, we may put 10-6+a-10.

In the first of these expressions, b is subtracted from a. In the other, b is subtracted from 10, the difference is added to a, and 10 is afterwards taken from the sum.

The two expressions are equivalent, because they consist of the same terms, with the addition, in one of them, of 10-10=0. The alteration is, in fact, nothing more than borrowing 10, for the sake of convenience, and then rejecting it in the result.

Instead of 10, we may borrow, as occasion requires, 100, 1000, &c.

Thus a-b=100-ta-100=1000-6+a-1000, &c.

54. The DIFFERENCE between a given number and 10, or 100, or 1000, d-c. is called the ARITHMETICAL COMPLEMENT of that number.

The arithmetical complement of a number consisting of one integral figure, either with or without decimals, is found, by subtracting the number from 10. If there are two integral figures, they are subtracted from 100; if three, from 1000, &c.

Thus the arithmetical compl’t of 3.46 is 10 – 3.46=6.54

of 34.6 is 100 – 34.6=65.4 of 346. is 1000-346,=654. &c.

According to the rule for subtraction in arithmetic, any number is subtracted from 10, 100, 1000, &c. by beginning on the right hand, and taking each figure from 10, after increasing all except the first, by carrying 1.

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The difference, or arith'l compl’t is 2.36875, which is obtained, by taking 5 from 10, 3 from 10, 2 from 10, 4 from 10, 7 from 10, and 8 from 10; we may take it without being increased, from 9.

Thus 2 from 9 is the same as 3 from 10,

3 from 9, the same as 4 from 10, &c.


55. To obtain the ariTHMETICAL COMPLEMENT of a number, subtract the right hand significant figure from 10, and each of the other figures from 9. If, however, there are ciphers on the right hand of all the significant figures, they are to be set down without alteration.

In taking the arithmetical complement of a logarithm, if the index is negative, it must be added to 9; for adding a negative quantity is the same as subtracting a positive one. (Alg. 81.) The difference between — 3 and +9, is not 6, but 12.

The arithmetical complement

of 6.24897 is 3.75103
of 2.98643 7.01357
of 0,62430 9.37570

of 2.70649 is 11.29351
of 3.64200

of 9.35001 0.64999

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