Let $q \in (0, 1)$ be a D5407: Positive real number such that

(i)	$f : [0, \infty) \to [0, \infty)$ is an D4367: Unsigned real function
(ii)	\begin{equation} f(x) = q^x \end{equation}

Let $b \in (0, \infty)$ be a positive real number. Using R3184: Second fundamental theorem of Riemann integral calculus, we have \begin{equation} \int^b_0 q^t \, d t = \left[ q^t \log q \right]^b_0 = q^b \log q - q^0 \log q = q^b \log q - \log q \end{equation} Since $q^b \to 0$ as $b \to \infty$, then \begin{equation} \int^{\infty}_0 q^t \, d t = - \log q \end{equation} $\square$