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. $\frac{1}{2} \left( \log_5 x + \log_5 y \right) - 2 \log_5 (x + 1)$

Verified step by step guidance

1. Start by applying the distributive property to the expression: multiply \( \frac{1}{2} \) by each term inside the parentheses. This gives \( \frac{1}{2} \log_5 x + \frac{1}{2} \log_5 y - 2 \log_5 (x + 1) \).
2. Use the power rule of logarithms, which states that \( a \log_b M = \log_b (M^a) \), to rewrite each term with coefficients as exponents inside the logarithms. So, \( \frac{1}{2} \log_5 x = \log_5 (x^{\frac{1}{2}}) \), \( \frac{1}{2} \log_5 y = \log_5 (y^{\frac{1}{2}}) \), and \( -2 \log_5 (x + 1) = \log_5 ((x + 1)^{-2}) \).
3. Now, rewrite the expression as the sum and difference of logarithms: \( \log_5 (x^{\frac{1}{2}}) + \log_5 (y^{\frac{1}{2}}) + \log_5 ((x + 1)^{-2}) \).
4. Apply the product rule of logarithms, which states \( \log_b A + \log_b B = \log_b (AB) \), to combine the first two terms: \( \log_5 (x^{\frac{1}{2}} y^{\frac{1}{2}}) + \log_5 ((x + 1)^{-2}) \).
5. Finally, use the product rule again to combine all terms into a single logarithm: \( \log_5 \left( x^{\frac{1}{2}} y^{\frac{1}{2}} (x + 1)^{-2} \right) \). This is the condensed form with coefficient 1.

Key Concepts

Properties of Logarithms

Properties of logarithms include rules such as the product rule (log_b(m) + log_b(n) = log_b(mn)), the quotient rule, and the power rule (a·log_b(m) = log_b(m^a)). These allow combining or breaking down logarithmic expressions to simplify or condense them into a single logarithm.

Condensing Logarithmic Expressions

Condensing logarithmic expressions means rewriting multiple logarithms as one single logarithm. This involves applying the product, quotient, and power rules to combine terms, ensuring the final expression has a coefficient of 1 in front of the logarithm.

Evaluating Logarithms Without a Calculator

Evaluating logarithms without a calculator requires recognizing values that simplify to known logarithmic results, such as log_b(b) = 1 or log_b(1) = 0. Using properties to rewrite expressions can help identify these values and simplify the expression further.