Help with inverse Function?

Question

When a camera flash goes off, the batteries immediately begin to recharge the flash's capacitor, which stores electric charge given by the following. (The maximum charge capacity is M and t is measured in seconds.)
Q(t) = M(1 - e^-t/a)
(a) Find the inverse of this function and explain its meaning.
t(Q) =

Q(t)=M(1-e^-t/a)
(Q/M)=-e^-t/a
ln(Q/m)=-(-t/a)
ok so now is it
ln(Q/M)=t/a (since it is -e^-t/a so is it -ln(e^-t/a) or is it ln(-e^-t/a)??

what about -a*ln(Q/M)-ln(1)

Answer 1

The inverse of Q(t) is a function y(t) such that Q(y(t)) = t. Therefore, M[1 - e^(-(y(t)/a)] = t , or t/M = 1 - e^(-y(t)/a) =
1 - 1/[e^(y(t)/a)]. Therefore, 1/[e^(y(t)/a) = 1 - t/M = (M - t)/M, or e^(y(t)/a) = M/(M - t), so y(t)/a = ln(M/(M - t)), and

y(t) = a ln(M/(M - t)).

Check: If this is the inverse, then we should have y(Q(t)) = t.
Now, y(Q(t)) = a ln (M/(M -Q(t))) = a ln (M/{M - M[1 - e^(-t/a)]}) =
a ln (1/[1 - {1 - e^(-t/a)}]) = a ln (1/e^(-t/a)) = a ln(e^(t/a) =
a*(t/a) = t .