Definition of an exponential function

If b is any number like that b = 0 and b ≠ 1 then an exponential function is function in the form,

                                                     f( x ) = b ^x

Where b is the base and x is any real number.

Notice that now the x is in the exponent & the base is a fixed number.  It is exactly the opposite through what we've illustrated to this point. To this point the base has been the variable, x in most of the cases, and the exponent was a fixed number.  Though, in spite of these differences these functions evaluate in precisely the similar way as those that we are utilized to. 

Before we get too far into this section we have to address the limitation on b. We ignore one and zero since in this case the function would be,

                             f( x ) = 0^x  = 0        and f( x) = 1^x  = 1

and these are constant functions & won't have several same properties that general exponential functions have.

Next, we ignore negative numbers so that we don't get any complex values out of the function evaluation.  For example if we allowed b = -4 the function would be,

                                   f(x)=(-4)x            ⇒ f (1/2)=(-4)(1/2)=√(-4)    

and as you can illustrates there are some function evaluations which will give complex numbers. We only desire real numbers to arise from function evaluation & so to ensure of this we need that b not be a negative number.

Now, let's take some graphs.  We will be capable to get most of the properties of exponential functions from these graphs.