Log1/2(2x-4)=-2 log3(x+1)+log3(x+3)=1

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To solve the equations provided, we will first simplify the given equations separately before combining them.

Given equations:
1) log1/2(2x-4) = -2
2) log3(x+1) + log3(x+3) = 1

Starting with equation 1:
Using the property of logarithms, we can rewrite log1/2(2x-4) = -2 as 1/2^(-2) = 2x - 4.
1/2^(-2) = 4, so we have:
4 = 2x - 4
2x = 8
x = 4

Next, we will simplify equation 2:
Using the property of logarithms, we can combine log3(x+1) + log3(x+3) as a single logarithm with multiplication:
log3((x+1)(x+3)) = 1
(x+1)(x+3) = 3
Expanding the left side, we get:
x^2 + 4x + 3 = 3
x^2 + 4x = 0
x(x+4) = 0
x = 0 or x = -4

Now, we will check these solutions in the original equations to see which ones are valid solutions:
1) For x = 4:
log1/2(2(4)-4) = -2
log1/2(4) = -2
log1/2(2^2) = -2
log1/2(4) = -2
-2 = -2 (true)

2) For x = 0:
log1/2(2(0)-4) = -2
log1/2(-4) = -2
(log(-4))/(log(1/2)) = -2
This is not a valid solution as the logarithm of a negative number is undefined.

3) For x = -4:
log1/2(2(-4)-4) = -2
log1/2(-12) = -2
This is not a valid solution as the logarithm of a negative number is undefined.

Therefore, the only valid solution to the given equations is x = 4.