Chom them logs!

We need L amount of logs to build a bridge! How much is that, you ask? I'll give you a hint:

\(\begin{cases} x^{ 2 }+xy+y^{ 2 }=a^{ 2 } \\ \log _{ \sqrt [ x ]{ a } }{ \sqrt { a } } +\log _{ \sqrt [ y ]{ b } }{ \sqrt { b } } =\frac { a }{ \sqrt { 3 } } \end{cases}\)

and

\(\frac{(x+y)}{2a}=\sqrt{\frac{L}{3}}\)

So, how many logs do we need?

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