Logarithms - Part I

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We have seen how either very large or very small numbers can be expressed as (base) index. When such large numbers have to be operated upon by addition, subtraction, multiplication or division, the laws of indices help us to simplify the calculations.  Another method to make calculations of multiplication and division easy, is the method of logarithms that we are going to study in this chapter. This method was developed by John Napier in 1614 A.D. and is in use today. Logarithm tables are available for references. Most modern scientific calculators have logarithmic functions for ease of use. For the present study, we will consider logarithms of positive real numbers only.

What we will study in this chapter :

1. Definition of Logarithm
2. Laws of Logarithm
3. How to determine logarithm of a given number
4. Antilogarithm
5. Applications

1. Definition of Logarithm
Logarithms of numbers are an extension of indices that we have studied in the last chapter.  As we go along in the chapter, the concept of logarithm of a number will become clear. Logarithm is also written in a short form as log. 

A number may be expressed as follows : 

number = (base) index  

then    

(index) = logarithm (base) (number)

It is read as, the (index) is log of  (number) to the (base).

For example, if we take the number = 1000 and base = 10,

Then 1000 = 103

3 = log 10 (1000)

That is, 3 is log of 1000 to the base 10.

Mathematically, the definition of logarithm of a number is given as : If a and b are positive real numbers 
( a 1), and b = ax, then x is called the logarithm of b to the base a.

x = log ab.

The condition a 1 is an important one, because if a = 1 then x can have any value. We have seen in the chapter of indices that 1 raised to any number will be 1 again!!  This means that if a = 1, then the log ab is not a definite number. This is not allowed. Hence for a valid logarithm or log of a number the condition a 1 has to be enforced.

From the definition of log of a number, notice the following :

  • a1 = a, log a a = 1; log of any number to the same base is 1.  

  • a0 = 1, log a 1 = 0; log of 1 to any base is zero.

2. Laws of Logarithm
There are four laws of logarithms that make operations such as multiplication and division of numbers easy.

1. Consider two numbers m and n which are positive real numbers. We have to calculate (m * n).

log a (m * n) = log a m + log a n                    [multiplication is converted into addition]

2. If we have to calculate  m/n

     then log a  (m/n)  =  log a m  -  log a n            [division is converted into subtraction]

3. log a ( mn )  = n log a m

4. log a m  =  log x m / log x a                             x 1 and x is a positive real number.

The fourth law gives the facility for the change of base. As mentioned earlier, logarithmic tables for standard bases are available. So if we are given any base, we can change the base to the standard bases with the above law and proceed with the calculations.

Numbers with base of 10 are called the ìcommon logarithmsî.  Tables for log to the base 10  or log 10 (or just plain log), are available easily. There is another base called ìeî. The logarithm to the base e is written as ln e (or just plain ln). They are called ìnatural logarithmsî. The value of e is 2.717. Natural logarithms have to be used for various scientific calculations.  For the current study, we are going to use only common lag table.

 

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