Proving $\lim_{n\to\infty} \left(1-2^{-\frac1n\log_2\frac n{\log n}}\right)^{\frac1{\log_2\frac{n}{\log n}}}=\frac12$

Can someone help me prove that this limit equals $1/2$:$$\lim_{n\to\infty} \left(1-\left(\frac{1}{2}\right)^{\frac{1}{n}\log_2(\frac{n}{\log(n)})}\right)^{\frac{1}{\log_2(\frac{n}{\log(n)})}}$$

I have tried taking the log, then applying l'Hospital's rule but I get another indeterminate form. I've also tried doing a Taylor expansion, but nothing works.