SOLUTION: Suppose {{{ 43^(3log (13,56)) = 56^(x*log (11,43)) }}}, where {{{x = log (b,n)}}}. Compute the pair of positive integers (b,n) that satisfies this equation, where b is the minimum

The starting equation is   = .    (1)


Take log base 11 of both sides.  
Using basic elementary properties of logarithms, you will get

     = .    (2)


Cancel common factor    in both sides.  You will get

     = .    (3)


Using   = , rewrite equation (3) in equivalent form

     = .     (4)


At this point, you have logarithms with common (the same) base 56.


Rewrite proportion (4) in equivalent form

     = .    (5)


According to the "base change" formula for logarithms, the right side of (5) is

     = .    (6)


So, proportion (5) is now this significantly simplified equation

     = .    (7)


Rewrite it equivalently in this form

    x = .


Again, apply the formula   =  to get

    x = .


But they want x =    (see the description of the problem).

It leads us to this equation

     = ,

which is the same as

     = .    (8)


From (8), we get the answer to the problem:  b = 13;  n =  = 1331.


ANSWER.  The solution to the problem is  b = 13;  n =  = 1331.