Equation solving involving logarithm

Question

I need help solving this equation to find the variable $RC$:

$$V_c=V_{in}\left(1-\frac{e^{-t}}{RC}\right).$$

I already know $V_c$, $V_{in}$ and $t$. I always get it wrong so the $RC$ is negative. It represents time and so it shouldn't be.

Answer 1

So, $$V_{c}=V_{in}\left(1-\exp\left({\frac{-t}{RC}}\right)\right)$$ Divide by $V_{in}$ and subtract $$\exp\left({\frac{-t}{RC}}\right)=1-\frac{V_{c}}{V_{in}}$$ Take logs $$\frac{-t}{RC}=\ln\left(1-\frac{V_{c}}{V_{in}}\right)$$ and re-arrange $$\frac{-t}{\ln\left(1-\frac{V_{c}}{V_{in}}\right)}=RC$$

Answer 2

If the equation is $$V_c = V_{in}\cdot\frac{1-e^{-t}}{RC},$$ then multiply the equation by $RC$ and divide it by $V_c$ to get

$$RC=V_{in}\cdot\frac{1-e^{-t}}{V_c}$$ which is negative if $e^{-t} > 1$, so when $t<0$.