Final answer:
The given logarithmic equation log4(x) + log4(x-2) = log4(15) can be rewritten using logarithmic properties as log4(x^2 - 2x). Equating the arguments of the logs, we get the quadratic equation x^2 - 2x - 15 = 0, which factors to (x-5)(x+3) = 0, yielding two solutions, x=5 and x=-3. As only positive arguments are valid for logarithms, x=5 is the correct solution.
Step-by-step explanation:
To solve for the variable in the logarithmic equation log₄(x) + log₄(x-2) = log₄(15), we can use the properties of logarithms.
Firstly, according to the properties of logarithms, when two logs with the same base are added, this is equivalent to the log of the product of their arguments. Therefore, we can combine the two logs on the left side: log₄(x) + log₄(x-2) = log₄(x*(x-2)). This simplifies to log₄(x^2 - 2x).
Now, because we have the same base and the logs are equal, the arguments must also be equal. That gives us the equation x^2 - 2x = 15.
We move all terms to one side to get the quadratic equation, x^2 - 2x - 15 = 0.
Next, we factor the quadratic: (x-5)(x+3) = 0. Thus, x must be 5 or -3.
However, we must check the solutions to ensure they work within the original logarithmic equation since log arguments must be positive. Inserting x=5 into the original equation, we find it is valid because log arguments are positive. Putting x=-3, the argument x-2 results in a negative value, which is not valid for logarithms. Thus, x = 5 is the only solution.1