if you are using $e^{rt}$, then since doubling time is (in your chosen unit of time) equal to $1$, we get $e^{(r)(1)}=2$. Taking logarithms to the base $e$, we get $r=\ln 2$.
It simplifies things to use the doubling time as the unit of time. However, in a more complicated situation, where we have say two kinds of cell, what unit shall we use? In the long run, the conventional units of time are more useful.
So the population at time $x$ is $e^{(\ln 2)(x)}$. Since $e^{\ln 2}=2$, this is just a fancy way of writing $2^x$.
Both models can be considered as continuous models, since $x$ is not restricted to be an integer. They are in fact the same model. And yes, we do get results that are not integers, but that is not important. No mathematical model will represent a complex biological phenomenon exactly. If the numbers involved are large, an "error" of half a cell has no practical significance. Indeed, the error is likely to be far larger than that.
The function $e^t$ has nice technical properties. For example, its derivative is $e^t$. the derivative of $2^t$ is the more messy $(\ln 2)2^t$. But anything one can do by using base $e$ can also be done using base $2$, or $10$. There will be small differences of detail, that's all.
For the sake of variety, without using template formulas, you can think in the following way. Exponential growth is about multiplying by some common growth factor every unit of time. We could call that growth factor $b$. So here, after two hours, we went from $125$ to $350$. that means $$125\cdot b^2=350$$ from which you can solve for $b$ and find that the growth factor is $\sqrt{\frac{350}{125}}$.
Now what was the original population? Well moving backwards in time two hours:
$$125\div\left(\sqrt{\frac{350}{125}}\right)^2=\frac{125^2}{350}$$ So the population at $t$ hours after the initial moment is $$P(t)=\frac{125^2}{350}\left(\sqrt{\frac{350}{125}}\right)^t$$
Best Answer
Hint. One may observe that $$ 1.029^{\color{red}{24}t}=\left(1.029^{\color{red}{24}}\right)^t\approx\left(1.986\right)^t. $$