Logarithm - Theory

Theory

Let a be  a positive real number , a ≠ 1 and a^x = m, then x is called the logarithm of m to the base a and is written as log a m, and conversely, if log a m=x, then a^x  = m 

Again, 
1. Logarithm to a negative base is not defined .
2. Logarithm of a negative number is not defined 

Two properties of logarithms: 

1. log a 1= 0 for all a >0, a ≠ 1 that is, log 1 to any base is zero 


2. log a a = 1 for all a>0, a=1 Log of a number to the same base is 1.

Laws of logarithms 

First law :
log (mn) = log m+ log n or log of product = sum of logs 

Second law :
 log (m/n) = log m - log n  log (m/n) = difference of logs 

Third law : 
log mⁿ = nlog m 

That is the power of n will come out of the log and will be in multiplication instead of being in exponent.

  Note : The first theorem converts a problem of multiplication into a problem of addition and the second theorem converts a problem of division into a problem of subtraction , which are far easier to perform than multiplication or division .That  is why logarithms are so useful in all numerical calculations.

Using above laws we can write in general: 

1. log (m n p) = log m+ log n + log p 

2. log (a1/a2/a3 .... ak) = log a1 + log  a2 + .... log ak

Common logarithm: Base 10 is assumed whenever it is not  indicated. therefore, we shall denote them by log m only. The logarithm calculated to base 10 are called common logarithms.