SOLUTION: Solve in integer numbers 2^(x+1) = 3^y+2 - 3^y.

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Question 1189533: Solve in integer numbers 2^(x+1) = 3^y+2 - 3^y.

Found 3 solutions by MathLover1, ikleyn, MathTherapy:
Answer by MathLover1(20850)   (Show Source): You can put this solution on YOUR website!

............note: and

.......simplify

.........take log of both sides











solution: set to zero

..........will be zero only if



.......simplify



so, solutions are
,


Answer by ikleyn(52914)   (Show Source): You can put this solution on YOUR website!
.
Solve in integer numbers 2^(x+1) = 3^y+2 - 3^y.
~~~~~~~~~~~~~~~~~ 

Left  side of the given equation is  2^(x+1).

Right side of the given equation is  9*3^y - 3^y = 8*3^y.


So, the given equation is

    2^(x+1) = 8*3^y.


Due to uniqueness of decomposition integer numbers into the product of prime numbers, 
from the last equation we conclude

    x + 1 = 3,  y = 0.


ANSWER.  x = 2;  y = 0.

Solved.

====================

Writing by @MathLover1 in her preceding post is irrelevant to the problem and can not be considered as a solution.

For your peace in mind, simply ignore her post.



Answer by MathTherapy(10557)   (Show Source): You can put this solution on YOUR website!

Solve in integer numbers 2^(x+1) = 3^y+2 - 3^y.
If this is , then we get: 

                       

               You can do the check!!
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