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This is more of mathematical problem, but I think there is an electrotechnical engineer that can help me.
So I was reading about I2C pull-up values in the Texas Instruments SLVA689 I2C Bus Pullup Resistor Calculation document and stumbled upon these equations and partial explanation:
What I don't understand is where does the yellow marked value of 0.8473 come from?
When you solve the eponential equation for times t1 and t2, i.e. how long it takes for the RC waveform to rise from 0 to 0.3×Vdd, and from 0 to 0.7×Vdd, then you can calculate the time how long it takes for the RC waveform to rise from 0.3×Vdd to 0.7xVdd, then you get ln(0.7)-ln(0.3) and that's 0.8473 and that is the coefficient for the RC rise time from 0.3 to 0.7.
