I have an assignment with the 555 IC in the astable mode.

I'm trying to demonstrate the tHIGH and tLOW formulas.

tHIGH =ln (2)*(Ra+Rb)*C

tLOW = ln (2)*Rb*C

Now, for the tLOW I'm using the discharge capacitor equation:

τ = Rb*C

vC = V0*e^(-t/τ)

⇔(1/3)*Vcc = (2/3)*Vcc*e^(-t/τ)

⇔1/2 = e^(-t/τ)

⇔ln (1/2) = -t/τ

⇔t = ln (2)*Rb*C

Now for the tHIGH I'm using the charge equation formula: vC = V0*(1-e^(-t/τ)), with τ = Ra + Rb

vC = V0*(1-e^(-t/τ))

⇔(2/3)*Vcc = (1/3)*Vcc*(1-e^(-t/τ))

⇔2 = 1 - e^(-t/τ)

⇔ln (1) = -(-t/τ)

but this is not looking correct... ln (1) = 0...

Where am I going wrong?