Questions: Write the logarithmic expression as a single logarithlogarithm with coefficient 1, and simplify as much as possible. Assume that the variables represent positive real numbers. 3 log u - 1/2 log v - 1/2 log w =

Solution Steps

To combine the given logarithmic expression into a single logarithm with a coefficient of 1, we can use the properties of logarithms. Specifically, we will use the power rule, which states that \( a \log b = \log b^a \), and the quotient rule, which states that \( \log a - \log b = \log \frac{a}{b} \). First, apply the power rule to each term, then combine them using the quotient rule.

We start with the expression given in the problem: \[ 3 \log u - \frac{1}{2} \log v - \frac{1}{2} \log w \]

Using the power rule of logarithms, we can rewrite each term: \[ 3 \log u = \log u^3 \] \[ -\frac{1}{2} \log v = \log v^{-\frac{1}{2}} = \log \frac{1}{\sqrt{v}} \] \[ -\frac{1}{2} \log w = \log w^{-\frac{1}{2}} = \log \frac{1}{\sqrt{w}} \]

Now we can combine these logarithmic expressions using the quotient rule: \[ \log u^3 - \log v^{\frac{1}{2}} - \log w^{\frac{1}{2}} = \log \left( \frac{u^3}{\sqrt{v} \cdot \sqrt{w}} \right) \]

Final Answer