A thermometer is placed in an oven preheated to a constant temperature of 390◦ F. Through a glass window in the oven door, an observer records that the thermometer reads 190◦ F after 1 minute and 230◦ F after 2 minutes. What is the initial reading of the thermometer?
I know that you have to use the formula $\frac{dT}{dt}=k(T-T_m)$. I don't know what to do next.
Newton's law of cooling would lead you to the differential equation,
$$\frac{dT}{dt} = k(T-T_i)$$
This is a very simple differential equation, which could be solved as:
$$ \frac{dT}{T-T_i} = k dt$$ $$ \Rightarrow \int_0^{t'} \frac{dT}{T-T_i} = \int_0^{t'} k dt$$ $$ \Rightarrow \ln\left(\frac{T}{T_i}\right) = k t'$$ $$ \Rightarrow T = T_i \exp (k t')$$
Now, the thing with exponential functions is, any point for which you know the values of $(t, T(t))$ can be taken as a starting point. I leave the rest to you as a simple exercise.