PROBLEM II.

To find the natural number corresponding to a given logarithm.

If four figures only be required in the answer, look in the table for the decimal part of the given logarithm, and if it cannot be found exactly, take the one nearest to it, whether greater or less; then the three figures in the first column, marked No. which are in a line with the logarithm found, together with the figure at the top of the table directly above it, will form the number required. Observing, that when the index of the given logarithm is affirmative, the integers in the number found must be one more than the number expressed by the index; but when the index of the given logarithm is negative, the number found will be wholly a decimal, and must have one cipher less placed between the decimal point and first significant figure on the left hand, than the number expressed by the index. Thus the natural number corresponding to the logarithm 2.90233 is 798.6, the natural number corresponding to the logarithm 3.77055 is 5896, and the natural number corresponding to the logarithm -3.36361 is .00231.

If the exact logarithm be found in the table, and the figures in the number corresponding do not exceed the index by one, annex ciphers to the right hand till they do. Thus the natural number corresponding to the logarithm 6.64068 is 4372000.

If six figures be required in the answer, find, in the table, the logarithm next less than the given one, and take out the four figures answering to it as before. Subtract this logarithm from the next greater in the table,

with two ciphers annexed to it, by the former; annex the quotient to the right hand of the four figures already found, and it will give the natural number required. Thus, let it be required to find the natural number corresponding to the logarithm 4.59859; then,

Divide 200 by 11, and the quotient will be 18, which annexed to the right hand of 3968, the four figures already found, makes 396818; therefore as the index is 4, the required natural number is 39681.8.

EXAMPLES.

1. Required the natural number answering to the logarithm 1.88030. Ans. 75.91.

2. Required the natural number answering to the logarithm 5.37081. Ans, 234861.

3. Required the natural number answering to the logarithm 3.11977. Ans. 1317.56.

4. Required the natural number answering to the lo garithm -2.97435. Ans. .094265.

PROBLEM III.

To multiply numbers, by means of logarithms.

CASE 1.-When all the factors are whole or mixed

RULE.

Add together the logarithms of the factors, and the sum will be the logarithm of the product.

EXAMPLES.

1. Required the product of 84 by 56.

Logarithm of 84 is

1.92428

3. Find by logarithms the product of 76.5 by 5.5. Ans. 420.75.

4. Find by logarithms the continued product of 42.35, 1.7364 and 1.76. Ans. 129.424.

CASE. 2.-When some or all of the factors are decimal numbers.

RULE.

Add the decimal parts of the logarithms as before, and if there be any to carry from the decimal part, add it to the affirmative index or indices, or else subtract it from the negative.

'Then add the indices together, when they are all of the same kind, that is all affirmative or all negative; but when they are of different kinds, take the difference between the sums of the affirmative and negative ones, and

Note. When the index is affirmative, it is not necessary to place any sign before it; but when it is negative, the sign must not be omitted.

EXAMPLES.

1. Required the continued product of 349.17, 25.43, .93521 and .00576.

In this example there is 2 to carry from the decimal part of the logarithms, which added to 3, the sum of the affirmative indices, makes 5; from this taking 4, the sum of the negative indices, the remainder is 1, which is the index of the sum of the logarithms, and is affirmative, because the sum of the affirmative indices together with the number carried, exceeds the sum of the negative indices.

2. Required the continued product of .0839, .7536, and .003179.

In this example there is 2 to carry from the decimal part of the logarithms, which subtracted from 6, the sum

c

of the negative indices, leaves 4, which is the index of the sum of the logarithms, and is negative, because the sum of the negative indices is the greater.

3. Required the continued product of 13.19, .3765, and .00415. Ans. .02061.

4. Required the continued product of 343, 1.794, 5.41 and .019. Ans. 63.25.

PROBLEM IV.

To divide numbers by means of logarithms.

CASE 1. When the dividend and divisor are both whole or mixed numbers.

RULE.

From the logarithm of the dividend, subtract the logarithm of the divisor, the remainder will be the logarithm of the quotient.

Note.-When the divisor exceeds the dividend, the question must be wrought by the rule given in the next

case.

EXAMPLES.

1. Required the quotient of 3450 divided by 23.

2. Required the quotient of 420.75 divided by 76.5.

Ans. 5.5.

3. Required the quotient of 37.1542 divided by 1.73958