Final answer:
To solve the logarithmic equation, log₅(x+1) - log₅(3x+15) = log₅((1)/(x)), we first combine the logs using the properties of logarithms, isolate x to form a quadratic equation, factor it, and find that the valid solution is x = 5.
Step-by-step explanation:
We are given the logarithmic equation log₅(x+1) - log₅(3x+15) = log₅((1)/(x)) and we want to solve for x. First, we will use the properties of logarithms to combine the logarithms on the left side. According to the properties of logarithms, log₅(A) - log₅(B) = log₅(A/B), so we get log₅((x+1)/(3x+15)) = log₅((1)/(x)).
Now, since the base of both logarithms is the same, we can equate the arguments (the expressions inside the logarithms) to get (x+1)/(3x+15) = 1/x. To solve this equation, we multiply both sides by x(3x+15) to clear the denominators, which gives us x(x+1) = 3x+15. Expanding and rearranging the equation gives us a quadratic equation: x^2 + x - 3x - 15 = 0, which simplifies to x^2 - 2x - 15 = 0.
Factoring the quadratic equation yields (x-5)(x+3) = 0, so the possible values for x are 5 and -3. However, if we substitute x = -3 back into the original equation, we would have a logarithm of a negative number, which is undefined. Therefore, the only valid solution is x = 5.