LOGARITHMS

                                                  LOGARITHMS

Introduction

By the use of logarithms it is possible to minimize the tedious arithmetical computations, which are found in some trigonometrical problems. The uses of logarithm help us to reduce multiplication and division to the simple work of addition and subtraction respectively; also raising a number to a power (involution) and extracting the roots (evolution) of a given number are reduced to the work of multiplication and division respectively.

Definition of Logarithms

Logarithm of a positive number of the power to which the base is to be raised to obtain the given number.

If a, x, N are three numbers, so that

                                                a^x = N, (N>0, a>0, a=! 1)…………….. (1)

Then the exponent x is the logarithm of N to the base a and this is written as

                                                x = logaN………………………………(2)

This relation is read as , x=logarithm of N to the base  a. To emphasize the idea, we repeat that, with respect to a given base, exponents are logarithms.

To remember this statement, the following relation may be noted.

                                                (base)^log= number.

It is clear, from definition, that the relations in (1) and (2) are equivalent i.e. two different ways of expressing the same statement. Being equivalent, any one form may be transformed to the other.

NOTE : If a^x=N then x=log N therefore  a^{log N} = N.    (where base is a)

Example

Let us consider the number 81.

Now, we have 81=3^4.

Therefore , By definition, 4=log₃81. Thus, for the same number 81, we get different logarithms for different bases. Hence, the base should be stated, otherwise the logarithm of a number is meaningless.

But, for brevity, the base is not indicated when base is understood to be known or if the relation is true for any value of the base.

Some Special Cases

1. The logarithm of 1, with respect to any finite nonzero base, is equal to zero.

Since   a^0=1. therefore log_a1=0.

2. The logarithm of any number, with respect to itself as the base, is equal to 1.

Since  a^1=a therefore log_aa=1.

Laws of Logarithms

If we remember how logarithms are close to indices, we may easily derive the laws of logarithms from the laws of indices.

1. The logarithm of the product of two numbers with respect to any base is equal to the sum of the logarithms of the numbers with respect to the same base., i.e.

Log_a(m x n) = log_am + log_an

2. The logarithm of the quotient of two numbers with respect to any base is equal to the difference of the logarithms of the dividend and the divisor with respect to the same base, i.e.

Log_a(m/n) = log_am - log_an

3. The logarithm of a power of a number with respect to any base is equal to the index of the power multiplied by the logarithm of the number with respect to the same base, i.e.

Log_am^n=n log_am