- Identify the domain of the equation as \(x > 0\).
- Simplify the equation using logarithmic properties to obtain \(\log_3(\frac{4}{x}) = 2\).
- Convert to exponential form, resulting in \(\frac{4}{x} = 9\).
- Solve for \(x\) to get \(x = \frac{4}{9}\), which matches option \(a\).
- The final answer is \(\boxed{a}\).
Explanation
- Analysis and Domain Identification
To solve the equation \(\log_{3}(4x) - 2\log_{3}x = 2\), we first define the domain of the variable \(x\). The argument of the logarithm must be positive, so \(4x > 0\) and \(x > 0\), which simplifies to \(x > 0\). We will use properties of logarithms to condense the left-hand side of the equation into a single logarithmic term.
- Condensing the Logarithmic Expression
We apply the power rule for logarithms, which states that \(k \log_b(m) = \log_b(m^k)\). Thus, \(2 \log_3(x) = \log_3(x^2)\). The equation becomes: \[\log_3(4x) - \log_3(x^2) = 2\] Applying the quotient rule for logarithms, \(\log_b(m) - \log_b(n) = \log_b\left(\frac{m}{n}\right)\), we get: \[\log_3\left(\frac{4x}{x^2}\right) = 2\] Simplifying the expression within the logarithm, noting that \(x > 0\), we have \(\frac{4x}{x^2} = \frac{4}{x}\). So, \(\log_3\left(\frac{4}{x}\right) = 2\).
- Solving for x and Selecting the Answer
Using the definition of a logarithm, \(\log_b(M) = K\) is equivalent to \(M = b^K\), we can rewrite the equation in its exponential form: \[\frac{4}{x} = 3^2\] \[\frac{4}{x} = 9\] Multiplying both sides by \(x\) and dividing by \(9\), we find \(x = \frac{4}{9}\). Since \(x = \frac{4}{9} > 0\), this solution is within the required domain. Comparing this result with the given options, we identify that the correct answer is \(a\).
Examples
Logarithmic equations like this are useful in fields such as acoustics to relate sound intensity levels to actual physical intensities, or in chemistry to calculate the pH of solutions based on hydrogen ion concentrations. Solving for the variable ensures we can accurately determine parameters like concentration or intensity given a measurement. Understanding how to condense and invert logarithms is a fundamental skill in modeling exponential growth and decay scenarios in real-world data analysis.