Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. (1/2)ln x - ln y

Verified step by step guidance

1. Identify the logarithmic expression given: \(\frac{1}{2} \ln x - \ln y\).
2. Recall the logarithmic property that allows you to move coefficients as exponents inside the logarithm: \(a \ln b = \ln b^{a}\).
3. Apply this property to the first term: \(\frac{1}{2} \ln x = \ln x^{\frac{1}{2}}\).
4. Use the logarithmic property for subtraction: \(\ln a - \ln b = \ln \left( \frac{a}{b} \right)\).
5. Combine the terms into a single logarithm: \(\ln \left( \frac{x^{\frac{1}{2}}}{y} \right)\).

Key Concepts

Properties of Logarithms

Properties of logarithms include rules such as the product, quotient, and power rules. These allow us to combine or break down logarithmic expressions. For example, the power rule states that a coefficient in front of a logarithm can be rewritten as an exponent inside the log.

Power Rule of Logarithms

The power rule states that a coefficient multiplied by a logarithm can be expressed as an exponent inside the logarithm: a * log_b(x) = log_b(x^a). This is essential for rewriting expressions like (1/2) ln x as ln(x^(1/2)) to help condense the expression.

Combining Logarithms Using Quotient Rule

The quotient rule states that the difference of two logarithms with the same base can be written as the logarithm of a quotient: log_b(A) - log_b(B) = log_b(A/B). This allows us to combine terms like ln(x^(1/2)) - ln(y) into a single logarithm ln(x^(1/2)/y).