SOLUTION: Suppose a and b are positive numbers for which {{{log(4,a)}}} = {{{log(10,b)}}} = {{{log(25,(a-b)).}}} What is the value of b/a?

Let x be the common value of ,  and .

So,  x = ,           (1)

     x = ,          (2)

     x = .      (3)


Then we can write

      = a               (4)

      = b              (5)

      = a-b            (6)


Multiply equation (4) by   (both sides).  You will get

      = ,  or  

      = .         (7)


Divide equation (7) by equation (5)  (side by side).  You will get

      = 

hence  

      = .              (8)


Square equation (8)  (both sides).  You will get

     =  = .    (9)


Now in the right side of (9) replace   by  (a-b), based on (6);

                             and replace   by "a", based on (4).


You will get then

      = ,

or  
      = .    (10)


We just are on the finish line and will celebrate a victory soon.


Now you see that (10) is a quadratic equation for .

To solve it quickly for , let's introduce new variable  = t.


Equation (10) takes the form

    t^2 = 1 - t,

or

    t^2 + t - 1 = 0.


Use the quadratic formula

     =  = .


Thus we have two solutions for t. But since the numbers "a" and "b" are positive (according to the condition),

we leave only one, positive solution:

    t =  = .


ANSWER.  There is one and only one solution for .   It is   = .