Exponential and Logarithmic Functions

Exponential function: if a positive real number  other than 1  , then the function f(x) defined by f(x) = ax for all x belong to R is called as exponential function.the domain of this exponential function is R .It is evident from its graph that it is everywhere continuos.
Graph of exponential function

Graph of y= ax , where 0

Graph of y= ax, where a>1

The shape of the graph y= ax for any a>1 is essentially the same as that of y = 2x.The graph of y = ax for a typical base b>1 and is rising (when viewed from left  to right ).for a base b where 0

Example of exponential function: y= 2x , y=3x+3
Logarithm function:


Si nce the exponential function y= ax for a >0 , a not equal to1  is montonic  (its graph is either  always rising for all x always falling ) , it must have an inverse that is itself monotonic .the inverse function  is called the logarithm of x to the base a .
Definition of logarithm function : If a > 0 and a  not equal to 1, the logarithm x to the base b is the function y=   that satisfy ay=x , that is y=   mean ay=x. Graph of logarithm function 

 Here green line represents graph of y=ax and blue line represent y=   .
Because y = bx is a continuos  , increasing function that satisfies ax >0 for all x ,   must also be continuos and  increasing  , and its graph must lies entirely  to the right of y-axis.