Write a power represented with a positive base and a positive exponent whose value is less than the base

You mean situation, when  a^b < a. 
There are two ways:

But exponent function a^b has to be increasing so a>1 
So first solution:  a>1 and 0<b<1.  Examples:


and second way:

Exponent function a^b has to be decreasing, so  0<a<1  
Second solution:  a<1 and b>1. Examples: