To find the number corresponding to a given Logarithm

If the Logarithm is found exactly, the corresponding number is taken. out at once; thus, seek for the logarithm in the Log. Tables and take out its numerical value in column "No."

To find the Arithmetical Complement of a Logarithm

The arithmetical complement of a logarithm is the number it wants of 10.000000; and the easiest way to find it is, beginning at the left hand, to subtract every figure from 9, except the last significant figure, which is to be taken from 10. Thus the arithmetical complement of 4:478309 is 5:521691 it is frequently used in the Rule of Proportion, and in trigonometrical calculations, to change subtraction into addition.

Accuracy of Logarithms

As regards the accuracy of logarithms to six places, they cannot be taken out correctly for numbers in excess of 435000, because the difference then ceases to change at the rate of 100 for I in the fourth figure.

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MULTIPLICATION BY LOGARITHMS

The logarithm of the product of any two numbers is equal to the sum of their logarithms.

Let n, n1, be any two numbers, l, l1, their logarithms.

Then log. 10 (n× n1) = l + l1

RULE. Add together the logarithms (Table of Logs.) of the two numbers to be multiplied and their sum will be a logarithm, the natural number corresponding to which will be the product required: if either the multiplicand or multiplier, or both of them, should consist wholly of decimals, and the index of the sum exceed 10, reject the 10, and the remainder will be the index of the logarithm answering to the product.

Note. In all these problems, if the resulting logarithm differs from a logarithm in the Tables by no more than 1, the number corresponding to the tabular logarithm may be generally taken as the correct answer; thus logarithm 3.763876 or 3.763878 gives natural No. 5806.

DIVISION BY LOGARITHMS

The logarithm of the quotient of any two numbers is equal to the difference of their logarithms.

Let n, n', be any two numbers, l, l', their logarithms.

RULE. From the logarithm of the dividend subtract the logarithm of the divisor, and the remainder will be a logarithm, whose corresponding number will be the quotient required. When the index of the divisor exceeds that of the dividend, borrow 10, and the remainder will be the index of the quotient.

Involution is the raising of powers from a given root.

When a number

is multiplied by itself, the product is called its second power, or square; when this product is multiplied by the given number, the last product is called its third power, or cube; and when the multiplication is again repeated, the fourth power, and so on. The first power, or number thus

raised, is called the root, and the number of the power to which the given number is raised, the index of that power: hence, to raise or involve a number to a given power, multiply its logarithm by the index of the power to which it is to be raised, and the product will be the logarithm of the power sought.

When the given number is a decimal fraction, reject the tens resulting from the multiplication of the index of the logarithm by the power.

Involution is expressed in the following manner: 10' means the square of 10, 103 means the cube of 10, 10' means the fourth power of 10, and

so on.

The logarithm of the power of any number is equal to the logarithm of the number multiplied by the index of the power.

Evolution is the extracting of the root of a given number, or finding a number which, when raised to the given power, will produce the given number it is consequently the reverse of involution, and is performed by dividing the logarithm of the number by the index of the power, and the quotient will be the logarithm of the root required.

When the given number is a decimal fraction, prefix to the index of its logarithm a figure less by one than the index of the root, and divide the whole by the index of the root. See the two examples of decimals below.

Evolution is expressed in the following manner. ✓10 means the square root of 10, 10 means the cube root of 10, and so on. It is also expressed in fractional indices, thus 10 means the square root of 10, 10 means the cube root of 10, and so on.

The logarithm of the root of any number is equal to the logarithm of the number divided by the index of the root.

RULE. Add together the logarithms of the quantities multiplied together, and subtract the logarithm of the divisor.

Or, add together the logarithm of the quantities multiplied and the arithmetical complement of the divisor.