Solve the equation `log_(2)(3-x)-log_(2)(("sin"(3pi)/4)/(5-x))=1/2+log_(2)(x+7)`

Explanation

To solve the equation \( \log_{2}(3-x) - \log_{2}\left(\frac{\sin\left(\frac{3\pi}{4}\right)}{5-x}\right) = \frac{1}{2} + \log_{2}(x+7) \), we will follow these steps:

### Step 1: Rewrite the equation using properties of logarithms
Using the property of logarithms that states \( \log_{a}(m) - \log_{a}(n) = \log_{a}\left(\frac{m}{n}\right) \), we can rewrite the left side of the equation:

\[ \log_{2}\left(\frac{3-x}{\frac{\sin\left(\frac{3\pi}{4}\right)}{5-x}}\right) = \frac{1}{2} + \log_{2}(x+7) \]
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