# Simplify RC Charging Equation

1. Jan 26, 2015

### Dr. Ficus

1. The problem statement, all variables and given/known data

Basically, I'm just trying to verify that I can do basic algebraic manipulation correctly. The objective: solve the charging equation of a simple RC circuit for (C), capacitance.

2. Relevant equations
$$v\left(t\right)={v}_{s}\left(1-{e}^{{-t}/{RC}}\right)$$
3. The attempt at a solution

I ended up with the following.
$$C=\frac{-t}{R\mathrm{ln}\left(\frac{{v}_{s}-{v}_{\left(t\right)}}{{v}_{s}}\right)}$$
Is this correct? Also, I have a suspicion that the log in the denominator can somehow be simplified more than just putting $$\mathrm{ln}\left(1-\frac{v\left(t\right)}{{v}_{s}}\right)$$ but I can't think of how. Thanks in advance!

2. Jan 26, 2015

### AssyriaQ

That is correct. You can, if you so prefer, use the principle
$$\ln(\frac{a}{b})=\ln a - \ln b$$