# Solving an equation for Tau

I have these two equations:

V1 = Vf(1-e^(-t1/tau)) and V2 = Vf=(1-e^(-t2/tau))

I need to solve for tau in terms of t1, t2, Vf, V1, and V2 but i cant seem to find a proper way to combine these equations.

can you help?

cepheid
Staff Emeritus
Gold Member
You can solve each equation individually for tau. Then, since the results must be equal, you can equate them. This will give you a relationship between V1, t1, and V2, t2.

Example of solving one of the equations:

$$V_1 = V_f (1 - e^{-t_1 / \tau} )$$​

Divide both sides by Vf:

$$\frac{V_1}{V_f} = 1 - e^{-t_1 / \tau}$$​

Subtract V1 / Vf from both sides. Then add the exponential term to both sides (in other words, rearrange the terms):

$$e^{-t_1 / \tau} = 1 - \frac{V_1}{V_f}$$​

Take the natural logarithm of both sides of the equation:

$$-\frac{t_1}{\tau} = \ln \left[1 - \frac{V_1}{V_f} \right]$$​

Solve for tau by multiplying both sides by tau and then dividing both sides by the ln term (in other words, cross-multiply):

$$\tau = -\frac{t_1}{\ln \left[1 - \frac{V_1}{V_f} \right]}$$​

Now, when you solve the second equation for tau, you'll get a similar answer in terms of V2 and t2. Since it must be true that tau = tau, you can equate these results.

erm, thanks, but i need tau in terms of all 5 variables, not just 3.

I shouldve made that clear; not hard to solve for tau using just 1 equation actually...

cepheid
Staff Emeritus
Gold Member
erm, thanks, but i need tau in terms of all 5 variables, not just 3.

I shouldve made that clear; not hard to solve for tau using just 1 equation actually...

Yeah, I know that. You should have read my post more closely. Do you agree with me, that using the exact same algebraic steps described in my first post, but applying them to the second equation (for V2), we will arrive at the following expression for tau?

$$\tau = -\frac{t_2}{\ln \left[1 - \frac{V_2}{V_f} \right]}$$​

If so, then it must be true that:

$$\tau = -\frac{t_2}{\ln \left[1 - \frac{V_2}{V_f} \right]} = -\frac{t_1}{\ln \left[1 - \frac{V_1}{V_f} \right]}$$​

EDIT: So basically V2, t2, V1, t1, can't all be independent variables. The equation above illustrates that they must be related to each other.

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yeah thats mathematically correct no doubt...but definitely not what i asked.

actually i already solved it myself just now. if you solve for the e pieces and divide one over the other, you can solve for tau in terms of all variables.

thanks though ceph