Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. (1/2)(log x + log y)

Verified step by step guidance

1. Start with the given expression: \(\frac{1}{2}(\log x + \log y)\).
2. Use the logarithm property that the sum of logarithms is the logarithm of the product: \(\log x + \log y = \log(xy)\).
3. Rewrite the expression as \(\frac{1}{2} \log(xy)\).
4. Apply the power rule of logarithms, which states that \(a \log b = \log(b^a)\), to move the coefficient inside the logarithm: \(\frac{1}{2} \log(xy) = \log((xy)^{\frac{1}{2}})\).
5. Express the final condensed form as a single logarithm: \(\log \sqrt{xy}\).

Key Concepts

Properties of Logarithms

Properties of logarithms include rules such as the product rule (log a + log b = log(ab)), the power rule (k log a = log(a^k)), and the quotient rule (log a - log b = log(a/b)). These rules allow combining or breaking down logarithmic expressions to simplify or condense them.

Condensing Logarithmic Expressions

Condensing logarithmic expressions means rewriting multiple logarithms as a single logarithm. This involves applying the properties of logarithms to combine sums, differences, and coefficients into one logarithmic term with coefficient 1, making the expression simpler and easier to evaluate.

Evaluating Logarithms Without a Calculator

Evaluating logarithms without a calculator requires recognizing logarithms of numbers that are powers of the base, such as log base 10 of 100 = 2. Simplifying expressions using properties can help rewrite logarithms in terms of known values, enabling exact evaluation without computational tools.