SOLUTION: are there any 2 different numbers that when switched with exponents are the same?

You are asking:

Are there any cases where  when A and B are
different numbers?

The first one that comes to mind is 



because both sides equal 16.

Another case (using fractions) is 



We prove that by proving that their logs are equal:

log of the left side



log of the right side



So 
because the log of the left side equals the log of the right side.

Other solutions are:





You can generalize this to:

 and 

then 

This can be proved the same way, by showing that the logs of the
left and right sides are equal.

In fact the first solution mentioned above



is the case of this when n=1

Edwin