Final answer:
To solve the given logarithmic equation, combine the logs using the property log(a) - log(b) = log(a/b), set the arguments equal to each other, solve for x, and check for extraneous solutions, ensuring they do not cause the arguments of the logs to be negative or zero.
Step-by-step explanation:
To solve the equation log (3x+1) - log (x-5)=log (x+4), first apply a logarithmic property that allows us to simplify the equation: The logarithm of a quotient is equal to the difference of the logarithms. This transforms our equation into log((3x+1)/(x-5))=log(x+4).
Since we have two logarithms equal to each other, we can deduce that their arguments (the numbers or expressions inside the log function) must be equal. Therefore, we can set the arguments of the logarithms equal to each other: (3x+1)/(x-5) = x+4.
Next, simplify the equation and solve for x. Cross multiply to get 3x+1 = (x+4)(x-5). Expanding the right side gives us a quadratic equation which we can then solve for x by setting the equation to zero and using the quadratic formula, if necessary.
After finding solutions for x, check for extraneous solutions by substituting the values back into the original logarithmic equation and ensuring that both sides of the equation remain equal.
Remember that the values of x cannot be such that you take the log of a negative number or zero, as those are outside the domain of the logarithm function. Hence, any solution leading to such a situation must be ruled out as an extraneous solution.