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 - 2x = 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)) = (x+1)(x+3) = Expanding the left side, we get x^2 + 4x + 3 = x^2 + 4x = x(x+4) = 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) = - log1/2(4) = - log1/2(2^2) = - log1/2(4) = - -2 = -2 (true)
2) For x = 0 log1/2(2(0)-4) = - log1/2(-4) = - (log(-4))/(log(1/2)) = - This is not a valid solution as the logarithm of a negative number is undefined.
3) For x = -4 log1/2(2(-4)-4) = - log1/2(-12) = - 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.
To solve the equations provided, we will first simplify the given equations separately before combining them.
Given equations
1) log1/2(2x-4) = -
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 -
2x =
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)) =
(x+1)(x+3) =
Expanding the left side, we get
x^2 + 4x + 3 =
x^2 + 4x =
x(x+4) =
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) = -
log1/2(4) = -
log1/2(2^2) = -
log1/2(4) = -
-2 = -2 (true)
2) For x = 0
log1/2(2(0)-4) = -
log1/2(-4) = -
(log(-4))/(log(1/2)) = -
This is not a valid solution as the logarithm of a negative number is undefined.
3) For x = -4
log1/2(2(-4)-4) = -
log1/2(-12) = -
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.