How many time constants will elapse before the charge on a capacitor falls to `0.1%`of its maximum value in a discharging RC circuit.

Explanation

In discharging `RC` circuit charge on capacitor
`q = q_(0)r^(-1//RC) = q_(0)e^(_t//tau)`
where `q_(0)` is maximum charge,
`tau` is time constant
When `q = 0.1 % of q_(0)`
`q= 0.1/100 q_(0) = 10^(-3) q_(0)`
`q= q_(0)e^(-t//tau)`
`10^(-3)q_(0) = q_(0)e^(-t//tau)`
`e^(t//tau) = 10^3`
`log_(e) e^(t//tau) = log_(e) 10^(3)`
`t/(tau) = log_(e) e = 2.303 log_(10010^(3) = 3 xx 2.3.3 log_(10) 10`
`t/(tau) = 6.909`
`t = 6.909 tau`.