Final answer:
To solve for x in the logarithmic equation, use the property of logs that combines terms into a single logarithm, then equate the inside of the logs and solve the resulting quadratic equation. The valid solution for x is 41.
Step-by-step explanation:
To find x in the equation log₂x + log₂(x - 5) = log₂(36x), you can use the properties of logarithms. Specifically, the property that log(a) + log(b) = log(ab) can simplify the left side of the equation. Thus, the equation becomes log₂(x(x - 5)) = log₂(36x). After simplifying, this reduces to log₂(x² - 5x) = log₂(36x).
By the property that if log₂A = log₂B, then A = B, we can deduce that x² - 5x = 36x. Solving this quadratic equation involves moving all terms to one side to get x² - 41x = 0. Factoring out x yields x(x - 41) = 0. Therefore, the solutions are x = 0 and x = 41. However, since log(0) is undefined, the only valid solution for x given the context of the question is x = 41.