Question

[Quick Answer Needed] Which of the following shows the extraneous solution to the logarithmic equation?

(Image Attached Below)

Answer 1

C

Explanation:

Given the logarithmic equation

`\log_4x+\log_4(x-3)=\log_4(-7x+21)`

First, notice that

`x>0\\ \\x-3>0\Rightarrow x>3\\ \\-7x+21>0\Rightarrow 7x<21\ x<3`

So, there is no possible solutions, all possible solutions will be extraneous.

Solve the equation:

`\log_4x+\log_4(x-3)=\log_4x(x-3),`

then

`\log_4x(x-3)=\log_4(-7x+21)\\ \\x(x-3)=-7x+21\\ \\x^2-3x+7x-21=0\\ \\x^2+4x-21=0\\ \\D=4^2-4\cdot 1\cdot (-21)=16+84=100\\ \\x_(1,2)=(-4\pm 10)/(2)=-7,\ 3`

Hence,
`x=3` and
`x=-7` are extraneous solutions