Final answer:
To solve the logarithmic equation log(x) = 1 − log(x − 3), a property of logarithms is used to combine the logarithms, and then the equation is transformed into a quadratic equation.option D is correct answer.
Step-by-step explanation:
The student is asking how to solve a logarithmic equation for the variable x. The equation is given by log(x) = 1 − log(x − 3). To solve this, we use the property of logarithms that states the logarithm of the quotient of two numbers is the difference between the logarithms of the individual numbers. Applying this property, we can rewrite the equation as:
log(x) + log(x − 3) = 1
Using the logarithm product rule, which combines the sum of two logarithms into the logarithm of the product of their arguments, the equation becomes:
log(x(x − 3)) = 1
To remove the logarithm and solve for x, we raise 10 to the power of both sides, because the base of the common logarithm is 10. Hence, we have:
x(x − 3) = 101
Simplifying and solving the quadratic equation gives us:
x2 − 3x − 10 = 0
Factoring the quadratic equation yields:
(x − 5)(x + 2) = 0
This results in two solutions for x: x = 5 and x = −2. However, by substituting these values back into the original equation, we find that x = 5 doesn't satisfy the original equation because it leads to log(0), which is undefined. Therefore, the only solution is x = −2, which corresponds to option (D).