Answer
$t = 0.693~\tau_L$
Work Step by Step
We can find an expression for $\frac{di}{dt}$:
$i = \frac{\mathscr{E}}{R}~(1-e^{-t/\tau_L})$
$\frac{di}{dt} = \frac{\mathscr{E}}{L}~e^{-t/\tau_L}$
We can find an expression for $\mathscr{E}_L$:
$\mathscr{E}_L = -L~\frac{di}{dt}$
$\mathscr{E}_L = -(L)~(\frac{\mathscr{E}}{L}~e^{-t/\tau_L})$
$\mathscr{E}_L = -\mathscr{E}~e^{-t/\tau_L}$
The magnitude of $\mathscr{E}_L$ is $~~\mathscr{E}~e^{-t/\tau_L}$
We can find $t$:
$\frac{\mathscr{E}_L}{\mathscr{E}} = 0.500$
$\frac{\mathscr{E}~e^{-t/\tau_L}}{\mathscr{E}} = 0.500$
$e^{-t/\tau_L} = 0.500$
$e^{t/\tau_L} = 2.00$
$\frac{t}{\tau_L} = ln(2.00)$
$t = ln(2.00)~\tau_L$
$t = 0.693~\tau_L$
