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. 2 log_b x + 3 log_b y

Verified step by step guidance

1. Identify the given expression: \$2 \log_{b} x + 3 \log_{b} y$.
2. Recall the logarithmic property that allows you to move coefficients as exponents inside the logarithm: \(a \log_{b} M = \log_{b} (M^{a})\).
3. Apply this property to each term: \$2 \log_{b} x = \log_{b} (x^{2})\( and \)3 \log_{b} y = \log_{b} (y^{3})$.
4. Use the logarithmic property for addition: \(\log_{b} A + \log_{b} B = \log_{b} (A \times B)\) to combine the two terms into a single logarithm.
5. Write the final condensed expression as \(\log_{b} (x^{2} y^{3})\), which is a single logarithm with coefficient 1.

Key Concepts

Properties of Logarithms

Properties of logarithms include rules such as the product, quotient, and power rules. These allow combining multiple logarithmic terms into a single logarithm by converting coefficients into exponents and combining sums or differences into products or quotients.

Power Rule of Logarithms

The power rule states that a coefficient in front of a logarithm can be rewritten as an exponent inside the logarithm, i.e., a·log_b(x) = log_b(x^a). This is essential for condensing expressions with coefficients into a single logarithm.

Condensing Logarithmic Expressions

Condensing logarithmic expressions means rewriting a sum or difference of logarithms as a single logarithm. This involves applying the product or quotient rules after using the power rule to handle coefficients, simplifying the expression into one logarithm with coefficient 1.