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Which statement about the potential solutions to [tex]\( 2 \log x - \log 3 = \log 3 \)[/tex] is true?

A. Both are extraneous solutions.
B. Only 3 is an extraneous solution.
C. Only -3 is an extraneous solution.
D. Neither is an extraneous solution.

Sagot :

To solve the equation [tex]\(2 \log x - \log 3 = \log 3\)[/tex], let's go through a detailed, step-by-step solution.

1. Given Equation:
[tex]\[ 2 \log x - \log 3 = \log 3 \][/tex]

2. Isolate the logarithmic expressions:
Add [tex]\(\log 3\)[/tex] to both sides of the equation:
[tex]\[ 2 \log x = 2 \log 3 \][/tex]

3. Simplify the equation:
Divide both sides by 2:
[tex]\[ \log x = \log 3 \][/tex]

4. Remove the logarithm by exponentiating both sides:
When the logarithms are equal, their arguments must also be equal:
[tex]\[ x = 3 \][/tex]

5. Check for extraneous solutions:
Logarithmic functions are only defined for positive arguments. This means that [tex]\( \log x \)[/tex] is only defined for [tex]\( x > 0 \)[/tex]. Hence, [tex]\( x = -3 \)[/tex] would be an extraneous solution since the logarithm of a negative number is undefined.

Therefore, the solution to [tex]\(2 \log x - \log 3 = \log 3\)[/tex] is [tex]\( x = 3 \)[/tex]. The solution [tex]\( x = -3 \)[/tex] is considered but rejected because logarithms are not defined for negative numbers.

So the true statement is:
"Only -3 is an extraneous solution."