Solve the equation: `2log_(3)x+log_(3)(x^(2)-3)=log_(3)0.5+5^(log_(5)(log_(3)8)`

Explanation

To solve the equation \( 2\log_{3}x + \log_{3}(x^{2}-3) = \log_{3}(0.5) + 5^{\log_{5}(\log_{3}8)} \), we will follow these steps:

### Step 1: Simplify the Right Side
First, we simplify the term \( 5^{\log_{5}(\log_{3}8)} \). By the property of logarithms, \( a^{\log_{a}(b)} = b \). Hence, we have:
\[ 5^{\log_{5}(\log_{3}8)} = \log_{3}8 \]
So, the equation becomes:
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