To solve the logarithmic equation log2(6-x) = 3 - log2x, we can use the properties of logarithms.
First, let's simplify the equation by combining the logarithms on the right side:
log2(6-x) + log2x = 3
Next, we can use the logarithmic product rule, which states that logb(M * N) = logb(M) + logb(N), to combine the logarithms on the left side:
log2[(6-x) * x] = 3
To solve for x, we can rewrite the equation using exponential form. Since the base of the logarithm is 2, we can rewrite the equation as:
2^3 = (6-x) * x
Simplifying the left side gives us:
8 = (6-x) * x
Now, we have a quadratic equation. Let's expand the right side:
8 = 6x - x^2
Re-arranging the equation gives us:
x^2 - 6x + 8 = 0
To solve this quadratic equation, we can factor or use the quadratic formula. Factoring the equation gives us:
(x - 4)(x - 2) = 0
Setting each factor equal to zero gives us two possible solutions:
x - 4 = 0 or x - 2 = 0
Solving these equations gives us:
x = 4 or x = 2
Therefore, the solutions to the logarithmic equation log2(6-x) = 3 - log2x are:
x = 4 or x = 2