Answer
$E_F \approx \frac{1}{2} (E_c + E_v) + \frac{1}{2} kTln [\frac{N_v}{Nc}]$
Work Step by Step
Here, $E_c - E_F/kT \gt \gt 1$ and $-(E_V - E_F)/kT \gt \gt 1$
So
$ \frac{N_v}{e^{-(E_V - E_F)/kT} } \approx \frac{N_c}{e^{E_c - E_F/kT} } $
Hence
$e^{(E_V - E_c + 2E_F)/kT} \approx \frac{N_v}{Nc} $
Here we solve for $E_F$
$E_F \approx \frac{1}{2} (E_c + E_v) + \frac{1}{2} kTln [\frac{N_v}{Nc}]$
