We have seen in the last article the nature of fractional exponents. Thus the square root of a5 equals a5/2 which may be put a2 1/2. In this way we may have an exponent of any fraction whatever, as a17/19. Between the exponents 2 and 3, we may have any number of fractional exponents all less than 3 and more than 2. So, also, the same between 3 and 4, or any other two consecutive numbers.

The consideration of fractional exponents or indices has led to the making of a series of decimal numbers called logarithms, which are treated in the manner in which exponents are treated; namely -

To multiply numbers add their logarithms.

To divide numbers, subtract the logarithm of the divisor from the logarithm of the dividend. -

To raise any number to a given power, multiply its logarithm by the exponent of that power.

To obtain the root of any power, divide the logarithm of the given number by the exponent of the given power.

As an example by which to exemplify the use of logarithms: What is the product of 25 by 375?

We first make this statement:

 Log. of 25 . = 1 ,, 375. = 2

Putting at the left of the decimal point the integer characteristic, or whole number of the logarithm at one less than the number of figures in the given number at the left of its decimal point.

To find the decimal part of the required logarithm we seek in a book of Logarithms (such as that of Law's, in Weale's Series, London) in the column of numbers for the given number 25, or 250 (which is the same as to the mantissa) and opposite to this and in the next column we find 7940 and a place for two other figures, which a few lines above are seen to be 39; annex these and the whole number is 0.397940. These we place as below:

Log. of 25 . = 1.397940.

Now, to find the logarithm of 375, the other factor, we turn to 375 in the column of numbers and find the figures opposite to it, 4031, which are to be preceded by 57, the two figures found a few lines above, making the whole, .574031, which are placed as below, and added together.

 Log. of 25. = 1.39794 ,, 375. =- 2.57403 The sum = 3.97197

This sum is the logarithm of the product. To find the product, we seek in the column of logarithms, headed o-, for .971971, the decimal part. We find first 97, the first two figures, and a little below seeking for 1971, the remaining four figures, we find 1740, those which are the next less, and opposite these, to the left, we find 7, and above 93, or together, 937; these are the first three figures of the required product.

### Examples Of Logarithms

For the fourth figure we seek in the horizontal column opposite 7 and 1740 for 1971, the remaining four figures of the logarithm, and find them in the column headed 5.

This figure 5 is the fourth of the product and completes it, as there are only four figures required when the integer number of the logarithm is 3. The completed statement therefore is -

 Log. of 25. = 1.397940, ,, ,, 375. = 2.574031, ,, ,, 9375 = 3.971971.

Another example in the use of logarithms. What is the product of 3957 by 94360?

The preliminary statement, as explained in last article, is -

 Log. 3957 = 3 ,, 94360 = 4

In the book of logarithms seek in the column of numbers for 3957. In the first column we find only 395, and opposite to this, in the next column, we find a blank for two figures, above which are found 59. Take these two figures as the first two of the mantissa, or decimal part of the required logarithm, thus, 0.59. Again, opposite 395 and in the column headed by 7 (the fourth figure of the given number), we have the four figures 7366. These are to be annexed to (0. 59) the first two obtained. The decimal part of the logarithm, therefore, is 0.597366.

To obtain the logarithm for 94360, the other given number, we proceed in a similar manner, and, opposite 943, we find 0.97; then, opposite 943 and in column headed 6, we find 4788, or, together, the logarithm is 0.974788. The whole is now stated thus -

 Log. of 3957 = 3.59737 ,, ,, 94360 = 4.97479 ,, ,, 373382000 = 8.57215 = sum of logs.

The two logarithms are here added together, and their sum is the logarithm of the product of the two given factors. The number corresponding to the above resultant logarithm may be found thus: Look in the column headed o for 57, the first two numbers of the mantissa, then in the same column, farther down, seek 2154, the other four figures of the mantissa; or, the four (1709) which are the next less than the four sought, and opposite these to the left, in the column of numbers, will be found 373, the first three figures of the product; opposite these, to the right, seek the four figures next less than 2154, the other four figures of the mantissa. These are found in the column headed 3 and are 2058. The 3 at the head of the column is the fourth figure in the product. From 2154, the last four figures of the mantissa, deduct the above 2058, or -

2154, 2058,

Remainder, 96.

At the bottom of the page, opposite the next less number (3727) to that contained in 3733, the answer already found, seek the number next less to the above remainder, 96. This is 92.8, and is in the column headed 8. Then 8 is the next number in the product. From 96 deduct 92.8, and multiply it by 10, or -

96

92.8

3.2 x 10 =32.

Then, in the same horizontal column, seek for 32 or its next less number. This is 23.2, found in column 2. This 2 is the next figure in the product. Additional figures may be obtained by the table of proportional parts, but they cannot be depended upon for accuracy beyond two or three figures. We therefore arrest the process here.

### The Square Of A Binomial

The product requires one more figure than the integer of the logarithm indicates; as the integer is 8, there must be nine figures in the product. We have already six; to make the requisite number nine we annex three ciphers, giving the completed product -

3957 x 94360 = 373382000.

By actual multiplication we find that the true product in the last article is 373382520. In a book of logarithms, carried to seven places, the required result is found to be 373382500 which is more nearly exact.

The utility of logarithms is more apparent when there are more than two factors to be multiplied, as, in that case, the operation is performed all in one statement. Thus: What is the product of 3.75, 432.95, 1712, and 0.0327?

The statement is as follows:

 Log. 3.75 = 0.574031 432.95 = 2.63644 1712. = 3.2335 .0327 = 8.51455 Product =___ 90891. = 4.95852 16 5

Explanations of working are given more in detail in most of the books of logarithms.