Estimating altitude from atmospheric pressure

An alpinist starts from a mountain refuge located at an altitude of $1800\,\mathrm{m}$. At the refuge, a barometric altimeter measures an atmospheric pressure of $820\,\mathrm{hPa}$.

Later during the ascent, the measured pressure has dropped to $650\,\mathrm{hPa}$.

Atmospheric pressure at altitude $h$ is modeled by
$$P(h)=P_0e^{-kh},$$where $P_0>0$ and $k>0$ are constants.

1. Let $h_1$ and $h_2$ be two altitudes. Show that
   $$\frac{P(h_2)}{P(h_1)} =e^{-k(h_2-h_1)}.$$

2. Deduce that the difference in altitude between the two points is
   $$h_2-h_1 = \frac{1}{k} \ln\left(\frac{P(h_1)}{P(h_2)}\right).$$

3. Assume that
   $$k=9.81\times10^{-4}\,\mathrm{m}^{-1}.$$ Estimate the altitude reached by the alpinist when the altimeter reads $650\,\mathrm{hPa}$.

4. The alpinist continues climbing until the atmospheric pressure is half of its value at the refuge.

   Show that the corresponding gain in altitude does not depend on the initial pressure, and compute this gain.

5. More generally, for $r\in(0,1)$, let $D(r)$ denote the gain in altitude required for atmospheric pressure to be multiplied by $r$.

   Show that, for all $r,s\in(0,1)$,
   $$D(rs)=D(r)+D(s).$$

   Identify the classical function underlying this property.