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) About Me  (Show Source):
You can put this solution on YOUR website!

2%5E%28x%2B1%29+=+3%5E%28y%2B2%29+-+3%5Ey............note:2%5E%28x%2B1%29=2%2A2%5Ex and 3%5E%28y%2B2%29+-3%5Ey=3%5Ey%2A3%5E2+-3%5Ey=3%5Ey%2A%283%5E2-1%29=8%2A3%5Ey

2%2A2%5Ex=+8%2A3%5Ey.......simplify

2%5Ex=+4%2A3%5Ey.........take log of both sides

log%28%282%5Ex%29%29=+log%28%284%2A3%5Ey%29%29

x%2Alog%28%282%29%29=+log%28%284%29%29%2Blog%28%283%5Ey%29%29

x%2Alog%28%282%29%29=+log%28%282%5E2%29%29%2By%2Alog%28%283%29%29

x%2Alog%28%282%29%29-+2log%28%282%29%29=y%2Alog%28%283%29%29

y=%28x%2Alog%28%282%29%29-+2log%28%282%29%29%29%2Flog%28%283%29%29

solution: set y to zero

0=%28x%2Alog%28%282%29%29-+2log%28%282%29%29%29%2Flog%28%283%29%29..........will be zero only if

0=x%2Alog%28%282%29%29-+2log%28%282%29%29

2log%28%282%29%29=x%2Alog%28%282%29%29.......simplify

2=x

so, solutions are
x+=+2, y+=+0


Answer by ikleyn(52914) About Me  (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) About Me  (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 matrix%281%2C3%2C+2%5E%28x+%2B+1%29%2C+%22=%22%2C+3%5E%28y+%2B+2%29+-+3%5Ey%29, then we get: 

                       

               You can do the check!!