Posted: Apr 25, 2016 5:07 pm
Here's how the pattern is progressing for positive integers. The table is colour coded and shows AB - BA.
The exceptions are clearly shown:
-A=1
-A=2 and B=3
-A=2 and B=4
(You can prove the pattern for most cases easily enough by assuming A>2 and B>A and then taking the differential with respect to B of AB - BA, keeping A constant, which is ln(A)AB - ABA-1 and noting that since B>A>1 we have ABA-1<BA and also noting that since A>2 and a natural number ln(A)AB>AB, hence the gradient is strictly positive and the function is monotonic on that region)
The exceptions are clearly shown:
-A=1
-A=2 and B=3
-A=2 and B=4
(You can prove the pattern for most cases easily enough by assuming A>2 and B>A and then taking the differential with respect to B of AB - BA, keeping A constant, which is ln(A)AB - ABA-1 and noting that since B>A>1 we have ABA-1<BA and also noting that since A>2 and a natural number ln(A)AB>AB, hence the gradient is strictly positive and the function is monotonic on that region)