Question

Which of the following is true regarding the solutions to the logarithmic equation below?

Answer 1

Hence, option D is correct.

x=9 is a true solution and x=-9 is a extraneous solution.

Explanation:

" In mathematics, an extraneous solution (or spurious solution) is a solution, such as that to an equation, that emerges from the process of solving the problem but is not a valid solution to the problem "

" A true solution is a valid solution of the equation "

We are given a equation as:

`2\log_(3)(x)=4\\\\log_(3)x^2=4\\\\x^2=3^4\\\\x^2=81\\\\x=9,-9`

Now as we know that the logarithmic function is not defined for negative 'x'.

Hence x= -9 is a invalid solution or we can say is a extraneous solution.

and x=9 is a true solution.

Hence, option D is correct.

x=9 is a true solution and x=-9 is a extraneous solution.

Answer 2

Note that the logarithmic function is determined for all
`x>0.`

Now solve the equation:

`2\log_3x=4,\\ \\\log_3x^2=4,\\ \\3^(\log_3x^2)=3^4,\\ \\x^2=81,\\ \\x_1=9,\ x_2=-9.`

Since
`x_2=-9<0,` this is an extraneous solution.

Since
`x_1=9>0,` this is true solution.

Answer: correct option is D