Final answer:
The equation log₂x + log₂(x+4) = 5 can be solved by combining the logarithms and rearranging the equation.
Step-by-step explanation:
To solve the equation log₂x + log₂(x+4) = 5, we can combine the logarithms using the property log(a) + log(b) = log(ab). So, we have log₂(x(x+4)) = 5. This can be rewritten as log₂(x² + 4x) = 5.
To solve for x, we can rewrite the equation in exponential form: 2⁵ = x² + 4x. Simplifying, we get 32 = x² + 4x. Rearranging, we have x² + 4x - 32 = 0.
From here, we can solve the quadratic equation using factoring, completing the square, or using the quadratic formula. The solutions to the equation are x = -8 and x = 4. However, since log₂x is not defined for negative values, the only valid solution is x = 4.