Prove some properties of a substance that decays at a rate proportional to the amount present

Consider a substance which disintegrates at a rate proportional to the amount present. Let $y = f(t)$ denote the amount of the substance present at time $t$ . Define the $1/n$ th life of the substance for a positive integer $n$ as the number $T$ such that

$f(T) = \frac{f(0)}{n}.$

Prove that the $1/n$ th life is the same for any given sample of the substance (i.e., prove that the $1/n$ th life is invariant under the starting size of the sample), and compute $T$ in terms of $n$ and the decay constant $k$ .
For given real numbers $a$ and $b$ , prove we can write the function $f(t)$ as
$f(t) = f(a)^{w(t)} f(b)^{1-w(t)}$

and determine the function $w(t)$ .

Proof. Since
$f(T) = \frac{f(0)}{n}$

we have

$\frac{f(T)}{f(0)} = \frac{1}{n} \quad \implies \quad \frac{f(0) e^{-kT}}{f(0)} = \frac{1}{n} \quad \implies \quad e^{-kT} = \frac{1}{n}.$

This is independent of the initial sample (since it depends only on the $1/n$ th life $T$ and the decay constant $k$ ). Further, computing the $1/n$ th life in terms of $n$ and the decay constant we have

$e^{-kT} = \frac{1}{n} \quad \implies \quad -kT = -\log n \quad \implies \quad T = \frac{\log n}{k}. \qquad \blacksquare$
Proof. Let
$w(t) = \frac{b-t}{b-a}.$

Then,

$\begin{align*} f(a)^{w(t)} f(b)^{1-w(t)} &= e^{-ka\left( \frac{b-t}{b-a} \right)} \cdot e^{-kb \left( 1 - \frac{b-t}{b-a} \right)} \\[9pt] &= \exp \left(- \frac{kab-kat}{b-a} - kb + \frac{kb^2-kbt}{b-a}} \right) \\[9pt] &= \exp \left( \left( \frac{1}{b-a} \right) \left( -kab +kat - kb^2 + kab + kb^2 -kbt \right) \right) \\[9pt] &= \exp \left( \left( \frac{1}{b-a} \right) (kat - kbt) \right) \\[9pt] &= e^{-kt} \\ & = f(t). \qquad \blacksquare \end{align*}$