If \(\log _{a} x=s\) and \(\log _{a} y=t,\) then \(\log _{a}(x y)^{2}=?\) A. 2\((s+t)\) B. \(s+t\) C. 4\(s t\) D. 2\(s t\) E. \(s t\)

Understanding the power rule of logarithms is crucial for simplifying logarithmic expressions, especially when dealing with exponents. The rule states that the logarithm of a number raised to an exponent is equal to the exponent times the logarithm of the number itself. In mathematical terms, this is represented as \( \log_{a}(b^c) = c \log_{a} b \).

For example, if you're faced with an expression like \( \log_{a}(x^3) \), the power rule allows you to transform it into \( 3 \log_{a}x \), which is often much easier to work with. Remember, this rule only applies when you have an exponent directly on the part of the expression inside the logarithm. It's a handy tool for simplifying and solving logarithmic equations, as it breaks down complex expressions into more manageable pieces.

Moving on to another fundamental concept, the product rule of logarithms is essential when logarithmic expressions involve multiplication. This rule helps us break down the log of a product into a sum of logs. Specifically, it states \( \log_{a}(b \cdot c) = \log_{a} b + \log_{a} c \).

For instance, if you encounter an expression like \( \log_{a}(xy) \), you can use the product rule to split it into \( \log_{a}x + \log_{a}y \). This separation makes the expression far more straightforward to deal with, whether you're on the path to solve for a variable or simplify further. The product rule is especially useful in combination with the power rule as they often work together to break down complex logarithmic expressions.

Last but not least, understanding logarithmic expressions as a whole is key to mastering the topic of logarithms. A logarithmic expression, at its core, represents the power to which a base number must be raised to obtain another number. The general form is \( \log_{a}b \), which asks for the exponent that turns 'a' into 'b'.

It's important to remember that the base 'a' is always positive and not equal to 1, and the argument 'b' must be positive. Logarithmic expressions can be manipulated and transformed using a variety of rules, such as the power and product rules discussed above, along with the quotient rule for division and the change of base formula.

When dealing with logarithmic equations, it's helpful to translate the operations of multiplication, division, and exponentiation into addition, subtraction, and multiplication, respectively, using logarithms. This makes the equations much more approachable and solvable, allowing for a clearer path to the solution.