This is called natural exponential function . It would be interesting to know if the inverse of the exponential function exists and has nice interpretation. This search motivates the following definition. Definition Let b > be a real number.
Then we say logarithm of a to base b is x if b x = a . Logarithm of a to base b is denoted by log b a . Thus log b a = x if b x = a . Let us work with a few explicit examples to get a feel for this.
We know = . In terms of logarithms, we may rewrite this as log = . Similarly, = 10000 is equivalent to saying log 10000 = . Also, = = is equivalent to saying log = or log = .
On a slightly more mature note, fixing a base b > , we may look at logarithm as a function from positive real numbers to all real numbers. This function, called the logarithmic function , is defined by log b : R + → R x → log b x = y if b y = x As before if the base b = , we say it is common logarithms and if b = e , then we say it is natural logarithms . Often natural logarithm is denoted by ln . In this chapter , log x denotes the logarithm function to base e , i.e., ln x will be written as simply log x .
The Fig . gives the plots of logarithm function to base , e and . Some of the important observations about the logarithm function to any base b > are listed below: Fig . Fig .
( ) We cannot make a meaningful definition of logarithm of non-positive numbers and hence the domain of log function is R + . ( ) The range of log function is the set of all real numbers. ( ) The point ( ,