Let's start with the given expression and understand how to simplify it step-by-step using properties of logarithms.
The given expression is:
[tex]\[ 4 \log_{\frac{1}{2}} w + \left(2 \log_{\frac{1}{2}} u - 3 \log_{\frac{1}{2}} v\right) \][/tex]
First, let's break this down and simplify each term individually:
1. Simplify each logarithmic term by using the property that \(a \log_b(x) = \log_b(x^a)\):
[tex]\[ 4 \log_{\frac{1}{2}} w = \log_{\frac{1}{2}} (w^4) \][/tex]
[tex]\[ 2 \log_{\frac{1}{2}} u = \log_{\frac{1}{2}} (u^2) \][/tex]
[tex]\[ -3 \log_{\frac{1}{2}} v = \log_{\frac{1}{2}} (v^{-3}) \][/tex]
Notice that \(\log_{\frac{1}{2}}(v^{-3})\) can be rewritten as \(- \log_{\frac{1}{2}}(v^3)\).
Now, let's combine these simplified logarithmic terms into a single logarithm:
2. Combine the logarithmic terms:
[tex]\[ \log_{\frac{1}{2}} (w^4) + \log_{\frac{1}{2}} (u^2) - \log_{\frac{1}{2}} (v^3) \][/tex]
Using the property of logarithms that says \(\log_b(x) + \log_b(y) = \log_b(x \cdot y)\) and \(\log_b(x) - \log_b(y) = \log_b(\frac{x}{y})\), we can combine these:
[tex]\[ \log_{\frac{1}{2}} (w^4) + \log_{\frac{1}{2}} (u^2) - \log_{\frac{1}{2}} (v^3) = \log_{\frac{1}{2}} \left( \frac{w^4 u^2}{v^3} \right) \][/tex]
Thus, we have rewritten the original expression as a single logarithm. Out of the provided options, this corresponds to:
[tex]\[ \boxed{\log_{\frac{1}{2}}\left(\frac{w^4}{u^2 v^3}\right)} \][/tex]
