# Question #07b7f

Mar 24, 2016

See steps below.

#### Explanation:

(a) Given expression is $V = {V}_{0} \cdot {e}^{- \frac{t}{R \cdot C}}$
To make $t$ its subject.
Divide both sides with ${V}_{0}$ and take natural logarithm of both sides.
In the first step we obtain
$\frac{V}{V} _ 0 = {e}^{- \frac{t}{R \cdot C}}$
In the second step
$\ln \left(\frac{V}{V} _ 0\right) = \ln {e}^{- \frac{t}{R \cdot C}}$, eliminating $e$
$\ln \left(\frac{V}{V} _ 0\right) = - \frac{t}{R \cdot C}$, now making $t$ its subject
$t = - \left(R \cdot C\right) \cdot \ln \left(\frac{V}{V} _ 0\right)$, absorbing $- v e \text{ sign in } \ln$ function
$t = R \cdot C \cdot \ln \left({V}_{0} / V\right)$.

(b) For the second part follow same steps as above to obtain
$t = \lambda \cdot \ln \left({N}_{0} / N\right)$