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

[tex]
\begin{aligned}
2 \log _6(x) & =2 \
\log _6(x^2) & =2 \
x^2 & =6^2 \
x^2 & =36 \
x & =6,-6
\end{aligned}
[/tex]

A. [tex]x=6[/tex] and [tex]x=-6[/tex] are true solutions
B. [tex]x=6[/tex] and [tex]x=-6[/tex] are extraneous solutions
C. [tex]x=6[/tex] is a true solution and [tex]x=-6[/tex] is an extraneous solution
D. [tex]x=6[/tex] is an extraneous solution and [tex]x=-6[/tex] is a true solution

Substitute x = 6 into the original equation and verify that it is a true solution.
Substitute x = − 6 into the original equation and identify it as an extraneous solution since the logarithm of a negative number is undefined.
Conclude that x = 6 is a true solution and x = − 6 is an extraneous solution.
The correct answer is: x = 6 is a true solution and x = − 6 is an extraneous solution ​

Explanation

Analyzing the Problem We are given the logarithmic equation $2

log _6(x) =2 an d t h es t e p s t oso l v e i t , w hi c h l e a d t o t h e p o t e n t ia l so l u t i o n s x=6 an d x=-6$. We need to determine which of these solutions are valid. Remember that the logarithm of a negative number is undefined in the real number system. Therefore, we must check each solution in the original equation.

Checking x=6 Let's check x = 6 by substituting it into the original equation:

2 l o g 6 ​ ( 6 ) = 2 c d o t 1 = 2
Since this is true, x = 6 is a valid solution.

Checking x=-6 Now let's check x = − 6 by substituting it into the original equation:

2 l o g 6 ​ ( − 6 )
Since we cannot take the logarithm of a negative number, x = − 6 is not a valid solution. It is an extraneous solution.

Conclusion Therefore, x = 6 is a true solution and x = − 6 is an extraneous solution.

Examples
Logarithmic equations are used in various fields, such as calculating the magnitude of earthquakes on the Richter scale, measuring sound intensity in decibels, and determining the pH of a solution in chemistry. Understanding how to solve logarithmic equations and identify extraneous solutions is crucial in these applications to ensure accurate and meaningful results. For example, in seismology, the Richter scale uses logarithms to quantify the energy released by an earthquake. If we incorrectly included a negative value in our calculations, we would arrive at a meaningless result, highlighting the importance of verifying solutions.

Answered by GinnyAnswer | 2025-07-05

The valid solution for the logarithmic equation is x = 6 , while x = − 6 is an extraneous solution due to being undefined in logarithmic terms. Thus, the correct option is C: x = 6 is a true solution and x = − 6 is an extraneous solution.
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Answered by Anonymous | 2025-07-07