What is the solution to the given equation?
$\log _6 4 x^2-\log _6 x=2$
The given equation is \(\log_6(4x^2) - \log_6(x) = 2\). First, we note the domain constraints for the equation to be defined. The arguments of the logarithmic functions must be positive: \(4x^2 > 0\) and \(x > 0\). These conditions imply that \(x\) must be a positive real number (\(x > 0\)).
Using the property of logarithms that \(\log_b(A) - \log_b(B) = \log_b\left(\frac{A}{B}\right)\), we can rewrite the left side of the equation as:
\[\log_6\left(\frac{4x^2}{x}\right) = 2\]
Simplifying the expression within the logarithm, since \(x
eq 0\), we have \(\frac{4x^2}{x} = 4x\).
The equation thus simplifies to:
\[\log_6(4x) = 2\]
By the definition of a logarithm, \(\log_b(y) = z\) is equivalent to \(b^z = y\). Applying this definition, we convert the logarithmic equation \(\log_6(4x) = 2\) to its exponential form:
\[4x = 6^2\]
Calculating \(6^2 = 36\), the equation becomes:
\[4x = 36\]
Dividing both sides by 4, we find:
\[x = \frac{36}{4} = 9\]
We have determined that \(x = 9\). Checking this against our domain constraint \(x > 0\), we confirm that \(x = 9\) is a valid solution. Comparing this result with the given choices, we see that it corresponds to option (a). Thus, the solution is \(x = 9\).
Logarithms are widely used in fields ranging from sound intensity measurements (decibels) to identifying the age of archaeological artifacts through radioactive decay. Understanding logarithmic equations allows you to solve for unknown variables in exponential models, which is essential for analyzing growth rates in population dynamics, finance, or chemistry. For example, if you know the final amount of money in a bank account and the interest rate, you can use logarithmic properties to calculate how many years it took for your initial deposit to grow to that amount.