Question

3
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 janet’s solutions.
of janet’s two solutions, \\(\boldsymbol{\text{drop - down menu}}\\) correct because \\(\boldsymbol{\text{drop - down menu}}\\).
drop - down menu options: only \\(x = - 2\\) is, only \\(x = 5\\) is, neither \\(x = 5\\) nor \\(x=-2\\) is, both \\(x = 5\\) and \\(x=-2\\) are

Explanation:

Step1: Recall log domain rule

For $\log(a)$, $a>0$ must hold.

Step2: Check $x=-2$ validity

Substitute $x=-2$: $\log(-2-3)=\log(-5)$ and $\log(-2)$ are undefined (arguments are negative).

Step3: Check $x=5$ validity

Substitute $x=5$: $\log(5-3)=\log(2)$ and $\log(5)$ are defined (arguments positive).

Step4: Verify $x=5$ in equation

Left-hand side: $\log(5-3)+\log(5)=\log(2)+\log(5)=\log(2\times5)=\log(10)=1$, which equals the right-hand side.

Answer:

only x = 5 is
$x=-2$ results in negative arguments for the logarithms, which are undefined, while $x=5$ satisfies the original equation and has positive arguments for all logarithms.