good exponential question

[Guest]

Exponential Issues - 5/26/03 Classic
x and y are positive integers 5^x-5^y=2^(y-1)5^(x-1) , what is the value of xy?
(A) 48
(B) 36
(C) 24
(D) 18
(E) 12


Answer E

(5^x-5^y)/5^(x-1)=2^(x-1)

5-5^(y-x+1)=2^(y-1)

5=2^(y-1)+5^(y-x+1)

y-x+1=0 for 5=2^(y-1)+5^(y-x+1)

and 2^(y-1)=4 ==> y=3 and x=4

xy=12

answer E

[raj]

how come

y-x+1=0 is assumed?

[StaceyKoprince]

Hi, please post the source for any questions you post - we are required to do this due to copyright issues. Instructors won't respond to any questions until the source is properly cited. Thanks!

[michael_shaunn]

The given question can be re-written as 5-5^[-(x-y)+1]=2^(y-1).(You can see it for yourself)
But before we move forward we need to have a look back at the original question.From the original question we can make out that right side of the question is always positive and thus,x>y in the LHS of the given question.

From the modified equation as mentioned above,x-y can't be greator than 1(else LHS will be a decimal and RHS an integer).
It can either be 0 or 1.Since x>y,x-y can't be equal to 0.Thus,x-y is equal to 1 which gives LHS as 4 and the modified equation transforns to 4=2^(y-1) giving y=3.From x-y=1 and y=3 we can get x=4.Hence x*y=12.

Hope it helps.

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