Question

select the correct answer from each drop - down menu.
janet solves this equation.
\\(\log(x - 3) + \log x = 1\\)
she finds two solutions, \\(x = 5\\) and \\(x = - 2\\).
complete this statement about janets solutions.
of janets two solutions, only \\(x = 5\\) is correct because
(options: both \\(x = - 2\\) and \\(x = 5\\) are extraneous solutions; both \\(x = - 2\\) and \\(x = 5\\) are valid solutions; \\(x = - 2\\) is an extraneous solution; \\(x = 5\\) is an extraneous solution)

Explanation:

Step1: Define log domain constraints

For $\log(a)$ to be defined, $a>0$. So:
$x-3>0 \implies x>3$, and $x>0$. Combined, $x>3$.

Step2: Check $x=-2$

$x=-2$ does not satisfy $x>3$, so $\log(-2-3)=\log(-5)$ and $\log(-2)$ are undefined.

Step3: Check $x=5$

$x=5>3$, so $\log(5-3)=\log(2)$ and $\log(5)$ are defined. Verify the equation:
$\log(5-3)+\log(5)=\log(2)+\log(5)=\log(2\times5)=\log(10)=1$, which matches the right-hand side.

Step4: Identify extraneous solution

$x=-2$ is extraneous because it violates the domain of the logarithm function.

Answer:

only x = 5 is correct because $\boldsymbol{x = -2}$ is an extraneous solution