Problem: In Newton's cooling experiment, the environment stays at constant temperature A and an object with heat constant k enters the environment at initial temperature $T_0 > A$. Derive a formula for the times when the object temperature drops $T_0/2$, $T_0/4$, and $T_0/8$.
I feel like I am missing something. Newton's cooling law is $T(t)=A+(T_0-A)e^{-kt}$ or $t = \frac{1}{k}\ln\left(\frac{T_0-A}{T-A}\right)$
So wouldn't the time at which the temperature drops $T_0/2$, $T_0/4$, and $T_0/8$ be $t = \frac{1}{k}\ln\left(\frac{T_0-A}{T_0/2-A}\right)$, $t = \frac{1}{k}\ln\left(\frac{T_0-A}{T_0/4-A}\right)$, and $t = \frac{1}{k}\ln\left(\frac{T_0-A}{T_0/8-A}\right)$ respectively?
Upon closer inspection, the problem does not seem to be asking for times when the temperature is $T_0/2$, $T_0/4$, and $T_0/8$. Instead, it seems to be asking for the temperature at times $T_0/2$, $3T_0/4$, and $7T_0/8$.
Therefore:
$t=\frac{1}{k}\ln\left(\frac{T_0-A}{T_0/2-A}\right)$
$t=\frac{1}{k}\ln\left(\frac{T_0-A}{3T_0/4-A}\right)$
$t=\frac{1}{k}\ln\left(\frac{T_0-A}{7T_0/8-A}\right)$