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Examples. 1. Multiply 512 by 256. 2. Multiply 8192 by 128. 3. Multiply 2048 by 256. 4. Divide 262,144 by 128. 5. Divide 1,048,576 by 512. 6. Divide 524,288 by 131,072. 7. What is the 2d power of 1024? 8. What is the 3d power of 64 ? 9. What is the 5th power of 16? 10. What is the 2nd root of 262,144 ? 11. What is the 3d root of 262,144 ? 12. What is the 4th root of 1,048,576 ? 13. What is the 5th root of 1,048,576 ? 14. What is the 6th root of 262,144 ?

The operations of multiplication, division, and the extraction of roots are very easy by means of this table. This table however contains but very few numbers. But an exponent of 2 may be found for all numbers-from 1 as high as we please. For 2' = 2, and 22 = 4. Hence the exponent of 2 answering to the number 3 will be between 1 and 2 ; that is, 1 and a fraction. So the exponents answering to 5, 6, and 7, will be 2 and a fraction, &c.

XLIX. A table may also be made of the powers of 3, or of 4, or any other number except 1, which shall have the same properties. Exponents might be found answering to every number from 1 upwards.

30 = 1, 3' = 3, 3 = 9, 3=27&c. The column of powers will always consist of the numbers 1, 2, 3, &c. but the column of exponents will be different according as the numbers are considered powers of a different number.

The formula at = y will apply to every table of this kind.

If any number except 1 be put in the place of a, and y be made successively 1, 2, 3, 4, a suitable value may be found for X, which shall answer the conditions.

If a be made 1, y will always be 1, whatever value be given to x; for all powers, as well as all roots of 1, are 1.

But if any number greater than 1 be put in the place of a, y may equal any number whatever, by giving x a suitable value

Giving a value to a then, we begin and make y successively 1, 2, 3, 4, &c. and these numbers will form the first column or columns of powers in the table. Then we find the values of x corresponding to these values of y, and write them in the second column against the values of y, and these form the column: of exponents. These exponents are called logariihms. The first column is usually called the column of numbers, and the second, the column of logarithms. The number put in the place of a, is called the base of the table. Whatever number is made base at first, must be continued through the table.

Observe that a = 1; therefore whatever base be used, the logarithm of 1 is zero. And 1 will be the logarithm of the base, for a' = d.

The most convenient number for the base, and the one generally used in the tables, is 10.

10° = 1, 10 = 10, 10% = 100, 10% = 1000, 10% = 10000, 10% = 100000, 10% = 1000000, &c.

Now to find the logarithm of 2, 3, 4, &c.
Make 10= 2, 10* = 3, 104 = 4, &c.

For all numbers between 1 and 10, x must be a fraction, because 10o = 1 and 10' = 10.

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As the process for finding the value of z in this equation is long and rather too difficult for young learners, we will suppose it already found.

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30103

Hence 101ööööö = 2 very nearly.

To understand this, we must suppose 10 raised to the 30103d power, and then the 100000th root of it taken, and this will differ very little from 2. The number .30103 is the logarithm of 2. The fractional part of logarithms is always expressed in decimals.

Having the logarithm of 2, we may find the logarithm of 4 by doubling it, for 2 =4. That of 8= 23 is found by tripling it, and so on.

The logarithm of 4 is .30103 X 2=.60206.
The logarithm of 8 is .30103 X 3 =.90309.
The logarithm of 16 is .30103 x 4 = 1.20412, &c.

4 771 2 1 3

Again 1010000000

= 3

very nearly.

Hence the logarithm of 3 is .4771213.

Since 2 X 3 = 6, the logarithm of 6 is found by adding the logarithm of 2 and 3 together.

.30103 +.4771213=.7781513 = logarithm of 6.

Since 39 = 9, the logarithm of 9 is found by multiplying that of 3 by 2. With the logarithms of 2 and 3 the logarithms of all the powers of each, and of all the multiples of the two may be found.

The logarithm of 5 may be found by subtracting that of 2 from that of 10, since 5 = . The logarithm of 10 is 1.

1—.30103 = .69897 = log. of 5. Now all the logarithms of all the multiples of 2, 3, 5, and 10 may be found. Hence it appears that it is necessary to find the logarithms of the prime numbers, or such as have no divisor except unity, by trial ; and then the logarithms of all the compound numbers may be found from them.

The decimal parts of the logarithms of 20, 30, &c. are the same as those of 2, 3, 4, &c. For, since the logarithm of 10 is 1 ; that of 100, 2; that of 1000, 3, &c., it is evident that add

ing these logarithms to the logarithms of any other numbers, will not alter the decimal part. Hence 1 added to the loga rithm of 2 forms that of 20, and 2 added to the logarithm of 2 forms that of 200, &c.

Log. 2 = .30103, log. 20 = 1.30103, log. 200 = 2.30103 log. 2000 = 3.30103.

The logarithm of 25 is 1.39794; that of 250 = 25 X 10 is 1 + 1.39794

2.39794; that of 2500 = 25 X 100 is 2 + 1.39794 = 3.39794.

The logarithms of all numbers below 10 are fractions, those of all the numbers between 10 and 100 are 1 and a fraction ; those of all numbers between 100 and 1000 are 2 and a fraction; those of all numbers between 1000 and 10000 are 3 and a fraction. That is, the whole number which precedes the fraction in the logarithm is always equal to the number of figures in the number less one. This whole number is called the index or characteristic of the logarithm. Thus in the logarithm 2.3576423, the figure 2 is the characteristic showing that it is the logarithm of a number consisting of three figures or between 100 and 1000.

As the characteristic may always be known by the number, and the number of figures in a number may be known by the characteristic, it is usual to omit the characteristic in the table, to save the room. It is useful to omit it too, because the same fractional part, with different characteristics, forms the loga rithms of several different numbers.

The logarithm of 37 is 1.568202. 37

101 = 3.7 = 10

10 The logarithm of 3.7 is .568202, which is the same as that of 37, with the exception of the index. 3762

103.575419 = 37.62 =

= 101.575419 100

1.568202

= 10.568202.

102

3762
= 3.762 =

103.876419

= 10-67541 1000

10

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That is, all numbers which are tenfold, the one of the other, have the same logarithm.

376200 has for its logarithm 5.575419.
37620

4.575419.
3762

3.575419. 376.2

2.575419. 37.62

1.575419. 3.762

0.575419. When a number consists of whole numbers and decimal parts, we find the fractional part of the logarithm in the same manner as if all the figures of the number belonged to the whole number, but we give it the index corresponding to the whole number only.

In most tables of logarithms they are carried as far as seven decimal places. Some however are only carried to five or six. The disposition of the tables is something different in different sets, but they are generally accompanied with an explanaion. When one set of tables is well understood, all others will be easily learned. The logarithms for the following examples may be found in any table of logarithms. They are used here as far as six places.

Examples

1. Multiply 43 by 25.

Find 43 in the column of numbers, and against it in the column of logarithms you will find 1.633468, and against 25 you will find 1.397940. Add these two logarithms together and their sum is the logarithm of the product. log. -43

1.633468 25

1.397940

1075

3.031408 Find this logarithm in the column of logarithms, and against it in the column of numbers you find 1075 which is the product of 43 multiplied by 25. The index, 3, shows that the number must consist of four places.

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