Calculating the base emitter voltage increase for an ideal transistor
The Veb increases 0.026 volts for every factor of 2.718.. (e) increase in current.

Or 0.018 volts for every factor of 2:1 increase; bandgap references use this fact, often hidden behind 0.036 v for 4:1 or 0.054 volts for 8:1.

Or 0.058 volts for every factor of 10:1. 

Or 0.58 volts for 10^10 increase, such as 1 picoAmp to 10milliAmp.