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The problem provided is:

\[
6^{2 \log_6 3} + \log_{\frac{1}{2}} 4 = \, ?
\]

To solve this, let's break it down into parts.

### Step 1: Simplify \(6^{2 \log_6 3}\)

Using the property \(a^{\log_a b} = b\), we can rewrite \(6^{2 \log_6 3}\) as follows:

\[
6^{2 \log_6 3} = (6^{\log_6 3})^2 = 3^2 = 9
\]

### Step 2: Simplify \(\log_{\frac{1}{2}} 4\)

Rewrite \(\log_{\frac{1}{2}} 4\) using the change of base formula. We know that \(4 = \left(\frac{1}{2}\right)^{-2}\), so:

\[
\log_{\frac{1}{2}} 4 = -2
\]

### Step 3: Combine Results

Now we substitute back:

\[
6^{2 \log_6 3} + \log_{\frac{1}{2}} 4 = 9 + (-2) = 7
\]

### Final Answer

\[
\boxed{7}
\]

Would you like more details or have any questions?

---

1. What is the general property of exponents in logarithmic expressions?
2. How does the base change affect logarithmic calculations?
3. Why does \(\log_{\frac{1}{2}} 4\) yield \(-2\)?
4. How is the expression \(a^{\log_a b}\) simplified?
5. Can this approach be applied to similar logarithmic problems?

**Tip:** Remember, converting expressions into known values using logarithmic properties can simplify complex expressions.

Обчисли: 6^{2 \log_6 3} + \log_{\frac{1}{2}} 4

Learn how to solve the logarithmic and exponential expression 6^(2 log6 3) + log(1/2) 4 with step-by-step explanations. This problem covers essential concepts like logarithmic identities and properties of exponents, suitable for Grades 9-11.

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