Question

Which of the following shows the extraneous solution to the logarithmic equation? log subscript 4 baseline (x) log subscript 4 baseline (x minus 3) = log subscript 4 baseline (negative 7 x 21) x = negative 7 x = negative 3 x = 3 and x = negative 7 x = 7 and x = negative 3

Answer 1

The extraneous solution is the value of the independent variable for which the problem has no solution. x=3 is an extraneous solution.

What is an extraneous solution?

An extraneous solution is a solution that develops from the process of addressing the problem but is not a real solution to the problem, such as an equation.

In order to know the extraneous solution, we need to find the value of x in the given equation,

$\rm log_4(x)+log_4(x-3)= log_4(-7x+21)$

Using the logarithmic property,

$\rm log_4[x(x-3)]= log_4(-7x+21)$

Taking the anti-log, we will get,

$x(x-3)= -7x+21\\\\x^2-3x=-7x+21\\\\x^2+7x-3x-21=0\\\\x(x+7)-3(x+7)=0\\\\(x-3)(x+7)=0$

If we substitute the factors with the 0, we will get the value of x as 3 and -7.

We know that extraneous solutions are those solutions, which when substituted in the initial equation, the equation does not return an equation.

x=3

$\rm log_4(x)+log_4(x-3)= log_4(-7x+21)$

$\rm log_4(x3)+log_4(3-3)= log_4(-21+21)\\\\\rm log_4(x)+log_4(0)= log_4(0)$

As the value of logₐ(0) is undefined, therefore, x=3 is an extraneous solution.

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