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Logarithmic expressions are a different way to express exponentiation. They answer the question: 'To what power must a certain base be raised to obtain a number?' For instance, the logarithmic expression \( \log_b x \) represents the power to which the base \( b \) must be raised to produce the number \( x \). When working with logarithmic expressions, it's crucial to understand their connection to exponents and how to manipulate them using various properties. These manipulations include simplifying expressions, condensing multiple logarithms into a single one, or expanding a single logarithm into multiple terms.
The expression \( \frac{1}{2}(\log x + \log y) \) involves adding two logarithms with the same base of 10 (implicit when no base is written) and multiplying by a fraction. Later steps will show how to utilize properties to condense these logs into a single, simpler expression. Paying close attention to the arithmetic and understanding each step of the process will assist you in mastering the handling of such expressions, allowing you to solve them with or without a calculator.

The power rule of logarithms is a transformative tool that simplifies expressions involving logarithms raised to an exponent. Essentially, the rule states that \( m \cdot \log_b a = \log_b (a^m) \). It allows you to move the coefficient of a logarithm inside the log as the exponent of its argument.
For example, when faced with \( \frac{1}{2}\log x + \frac{1}{2}\log y \) from the initial exercise, we can use the power rule to transform it into \( \log x^{\frac{1}{2}} + \log y^{\frac{1}{2}} \). This maneuver is not merely aesthetic—it sets up the expression for further simplification using other logarithmic properties and can also make it easier to evaluate numerical expressions.
To fully leverage the power of the power rule, one must remember to carefully handle the arguments and their exponents, sticking to proper mathematical syntax to avoid common pitfalls.

The product rule for logarithms is key when you're looking to combine two or more logarithms into a single expression. It states that the logarithm of a product is the sum of the logarithms: \( \log_b a + \log_b c = \log_b (ac) \). This property is incredibly useful for condensing complex logarithmic expressions. In our exercise, once we've applied the power rule, we end up with \( \log x^{\frac{1}{2}} + \log y^{\frac{1}{2}} \). The product rule enables us to merge these logs into \( \log (x^{\frac{1}{2}}y^{\frac{1}{2}}) \).
Further simplification results in \( \log (xy)^{\frac{1}{2}} \), a cleaner expression where the argument \( xy \) is under a single square root. Recognizing when to apply the product rule is vital—it can not only help to simplify expressions but is also crucial in solving logarithmic equations. Just make sure to apply it only to logarithms that have the same base and are being added together.