Does current drive voltage in a transistor?
Yes. That is a very close approximation, because the 10 volts is very large compared to a diode voltage range of 0.5 to 0.7 volts for 1mA (estimated).

Notice you can use 0.060 volts estimated change in Vbe for 10:1 or 1:10 change in base current. Thus you can add 0.06 volts, and estimate the voltage at 1,000uA base current. Or you can subtract 0.06 volts, and estimate the voltage at 10uA or 1uA or 0.1uA base current.

Circuit Simulators use a similar iteration, hidden inside a matrix computation.