Finding the Base-Emitter Voltage in an NPN Transistor
We have two equations for two unknown quantities (Vbe and Ic):

IC=beta(Vcc-Vbe)/RB  and Ic=Is*exp[(VBE/VT)-1].

(For normal operation in the active mode the exponential expression is much larger than "1" - hence, we can neglect the "1").


An exact solution is possible (graphical solution) if we plot both functions Ic=f(VBE). The point where both curves meet is the actual operating point (Ic and VBE).
An exact numerical solution in one step is not possible (because of the exponential function). However, the two following alternatives exist: 


(a) Iterative solution: Start with VBE=0.65 volts and verify - using both functions - if this value was too large or too small (and try a second run).

(b) Replace the exponential function by the first (two) part(s) of the corresponding power series: exp(x)=1 + x + x²/2! + ......This approximation allows a direct but approximate numerical solution (mathematical combination of both functions).