Questions: Write the expression as a single logarithm. 3 logb x - 1/2 logb z + 2 logb w

Solution Steps

The power rule of logarithms states that \( k \log_b a = \log_b a^k \). Apply this rule to each term in the expression: \[ 3 \log_b x = \log_b x^3, \quad -\frac{1}{2} \log_b z = \log_b z^{-\frac{1}{2}}, \quad 2 \log_b w = \log_b w^2. \] Thus, the expression becomes: \[ \log_b x^3 + \log_b z^{-\frac{1}{2}} + \log_b w^2. \]

The product rule of logarithms states that \( \log_b a + \log_b c = \log_b (a \cdot c) \). Apply this rule to combine the logarithms: \[ \log_b x^3 + \log_b z^{-\frac{1}{2}} + \log_b w^2 = \log_b \left( x^3 \cdot z^{-\frac{1}{2}} \cdot w^2 \right). \]

Simplify the expression inside the logarithm: \[ x^3 \cdot z^{-\frac{1}{2}} \cdot w^2 = \frac{x^3 w^2}{\sqrt{z}}. \] Thus, the final expression is: \[ \log_b \left( \frac{x^3 w^2}{\sqrt{z}} \right). \]

Final Answer