Question

csd pre - calculus cp ic 25 - 26 solving equations using properties of logarithms using the power rule which statement about the potential solutions to 2 log x−log 3 = log 3 is true? only 3 is an extraneous solution. neither is an extraneous solution. only −3 is an extraneous solution. both are extraneous solutions.

Explanation:

Step1: Use logarithm properties

First, use the power - rule of logarithms \(n\log_aM=\log_aM^n\) and the quotient - rule \(\log_aM-\log_aN = \log_a\frac{M}{N}\) on the left - hand side of the equation \(2\log x-\log3=\log3\).
\(2\log x-\log3=\log x^2-\log3=\log\frac{x^2}{3}\). So the equation becomes \(\log\frac{x^2}{3}=\log3\).

Step2: Remove the logarithms

Since the logarithms on both sides have the same base (assuming base 10 or any common base \(a>0,a
eq1\)), we can set the arguments equal: \(\frac{x^2}{3}=3\).

Step3: Solve for \(x\)

Cross - multiply to get \(x^2 = 9\), then \(x=\pm3\).

Step4: Check for extraneous solutions

The domain of the logarithmic function \(y = \log x\) is \(x>0\). When \(x = 3\), \(\log x\) is well - defined. When \(x=-3\), \(\log x=\log(-3)\) is not a real number in the set of real numbers. So \(x = - 3\) is an extraneous solution.

Answer:

Only \(-3\) is an extraneous solution.