In Exercises 41–70, 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. Recall the logarithmic property that allows you to move coefficients in front of logarithms as exponents inside the logarithm: $a\cdot \log b_{}x=\log b_{}{x}^a$.
2. Apply this property to each term in the expression: $2\; log\_b\; x\; =\; log\_b\; (x^2)$ and $3\; log\_b\; y\; =\; log\_b\; (y^3)$.
3. Use the logarithmic property that the sum of logarithms with the same base is the logarithm of the product: $log\_b\; A\; +\; log\_b\; B\; =\; log\_b\; (A\; imes\; B)$.
4. Combine the two logarithms into a single logarithm: $log\_b\; (x^2)\; +\; log\_b\; (y^3)\; =\; log\_b\; (x^2\; imes\; y^3)$.
5. Write the final condensed expression as a single logarithm with coefficient 1: $log\_b\; (x^2\; y^3)$.

Key Concepts

Properties of Logarithms

Properties of logarithms include rules such as the product, quotient, and power rules. These allow combining or breaking down logarithmic expressions. For example, the power rule states that a coefficient in front of a log can be rewritten as an exponent inside the log: a·log_b(x) = log_b(x^a).

Condensing Logarithmic Expressions

Condensing logarithmic expressions means rewriting multiple logarithms as a single logarithm. This is done by applying the product rule (log_b(M) + log_b(N) = log_b(M·N)) and the power rule to combine terms into one log with coefficient 1, simplifying the expression.

Evaluating Logarithms Without a Calculator

Evaluating logarithms without a calculator involves recognizing values that simplify to known logarithmic results, such as log_b(b) = 1 or log_b(1) = 0. Using properties and known bases helps simplify expressions to exact values instead of decimal approximations.