\( y = \log _ { 6 } ( 2 x + 1 ) - \log _ { 6 } ( x - 2 ) \)

To solve the equation \( y = \log_6(2x + 1) - \log_6(x - 2) \), we can use the properties of logarithms to combine the logs. Specifically, we use the quotient rule, which states that \( \log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right) \). Thus, we rewrite the equation as: \[ y = \log_6\left(\frac{2x + 1}{x - 2}\right) \] This means we are looking at the logarithm base 6 of the fraction \(\frac{2x + 1}{x - 2}\), indicating a relationship where the expression is defined provided \(x - 2 > 0 \Rightarrow x > 2\). Remember that logarithms are undefined for non-positive values, which shapes our working domain. Logarithmic equations often pop up in real-world scenarios, especially in fields like finance, computer science, and biology. For example, they can be used to model exponential growth or decay, such as population growth or the half-life of radioactive materials. Understanding how to manipulate logarithmic expressions can be quite handy in these contexts, so never underestimate their power! If you want to dig deeper into logarithms, there are plenty of math resources available online, including Khan Academy and Math is Fun. They break down the concepts into easily digestible pieces with interactive examples you can try out. It’s a fun way to level up your math skills while reinforcing your understanding of how logarithms work in various applications!