SOLUTION: let. a, b, c, d be positive integers and (log( b)/log (a))=3/2 ,(log (d)/log (c))=5/4 ,if a-c=9 then b - d = .....

, where0 a, b, c, d are positive integers.







Let ,,, and perhaps
p and q can be positive integers.  We will see.







Factor the left side:



If those expressions in parentheses can be positive integers,
they can only be the unequal pair of factors of 9, which are
1 and 9. The p-q² is smaller, so it would be 1 and p+q² is
larger so it would be 9:

, 



Add the equations:




Subtract the equations:





So we now see that p and q can be positive integers.

,,,

,,,

,,,





Edwin