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."
