BJT Base-Emitter Voltage
\$A\$ is the area. \$I_S\$ is the device's saturation current, which is proportional to \$A\$.  The device with area \$nA\$ would have saturation current \$n I_S\$. So your second line should have been:

$$ = V_T \ln \left( {{I_C}\over{I_S}} \right) - V_T \ln \left( {{I_C}\over{n I_S}} \right) $$

(Note \$n\$ in denominator instead of numerator) The rest then goes through to what you were trying to prove.