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

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[Quick Answer Needed] Which of the following shows the extraneous solution to the-example-1

1 Answer

4 votes

Answer:

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

User Juned Ahsan
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