Find quantity from price elasticity I have $P_1 = 24$, $Q_1 = 800000$, $P_2 = 32$ and price elasticity $e = -8$ and need to find the function $P(Q)$.

I guess I need to find $Q_2$ by

$$
\frac{(Q_2-Q_1)/Q_1}{-(P_2-P_1)/P_1} = -8 \Leftrightarrow Q_2 = \frac{8800000}{3}
$$

But does it make sense that the quantity will be so much higher by a price increment and thus not follow the law of demand? I know that $e = -8$ is a high price elasticity but this is quite extreme I think.
You have that price elasticity is defined by:

$$\varepsilon_{d} = -\frac{\partial Q}{\partial P}\frac{P}{Q} \approx -\frac{P}{Q}\frac{\Delta Q}{\Delta P}$$

Initially we have $P = P_{1}$ and $Q=Q_{1}$ and after the perturbation we have $\Delta P = (P_{2} - P_{1})$ and $\Delta Q = (Q_{2} - Q_{1})$, therefore:

$$-\frac{P_{1}}{Q_{1}}\frac{Q_{2}-Q_{1}}{P_{2}-P_{1}}=-8 \implies Q_{2}-Q_{1}=\frac{8Q_{1}(P_{2}-P_{1})}{P_{1}}$$

And therefore:

$$ Q_{2}=Q_{1}+\frac{8Q_{1}(P_{2}-P_{1})}{P_{1}}=\frac{8800000}{3} $$

This is so large, partly because the price elasticity is so large, but also probably because of the error in the finite difference approximation method over such large changes in $P$.