Rewriting a recurrence in another form

Question

I have the following recurrence:

$$T(n) = n^{\log_2(k)/\log_2(n)} T(n/2) + n$$

I've tried to rewrite $n^{log_2(k)/log_2(n)}$ in another form but I don't see another way except $\sqrt[\log_2(n)]{...}$. Does anyone see a clever format to that?

Answer 1

I've tried to rewrite n^{log_2(k)/log_2(n)} in another form but I don't see another way except square root of degree log_2(n). Does anyone see a clever format to that?

One may observe that $$ n^{\log_2(k)/\log_2(n)}=n^{\log(k)/\log(n)}=e^{\frac{\log(k)}{\log(n)} \cdot \log (n)}=k. $$