Servois' 1814 Essay on a New Method of Exposition of the Principles of Differential Calculus, with an English Translation - The Natural Logarithm Series

In Section 16, Servois applies his expansion formulas to $F(x) = \varphi ^x(z)$, where $\varphi$ is a distributive function that is commutative with constant factors. That is $\varphi$ and $z$ are given and what varies in $F(x)$ is the order of the function $\varphi$. It is clear that this means when $x$ is an integer, but Servois doesn’t tell us how to interpret $F(x)$ for other values of $x$. Letting $\alpha$ be the constant increment in $x$, Servois finds an expression for $\Delta^n F(x)$. By his definition of the differential in (39), it follows that $${\mbox d} F(x) = \Delta F(x) - \frac{1}{2}\Delta^{2}F(x) + \cdots\phantom{xxxxxxxxxxxxxx}$$ $$\phantom{xxxxxxxxxx}=\varphi^{x}\left[\left(\varphi^{\alpha} - 1\right)z - \frac{1}{2}\left(\varphi^{\alpha} - 1\right)^{2}z + \frac{1}{3}\left(\varphi^{\alpha} - 1\right)^{3}z - \cdots\right].$$

Servois calls the expression in the brackets the logarithm of $\varphi^{\alpha}$ of $z$. The analogy between this formula and the usual power series for $\ln (x)$ (as opposed to $\ln (1 + x)$) is clear. However, Servois observes that the operator $\ln \varphi^{\alpha}$ is distributive and commutes with the function $\varphi$ and the constant factor, which is certainly not the case for the ordinary natural logarithm.

Servois then derives analogs of the familiar properties of the ordinary logarithm for this operator. Then, by looking at the inverse of this logarithm, Servois is able to derive a power series in his (62) that has the form for the usual power series for $F(x) = e^x$. Although no where in these formula arguments does Servois address the issue of non-integer values of $x$ and $\alpha$, his most important application in Section 18 will be when the function $\varphi$ is the constant factor $a$, which is a distributive function that certainly commutes with constant factors. In this case, as Servois observes at the end of Section 17, $\ln \varphi^{\alpha} z$ is the natural logarithm of $a^{\alpha} z$ and the inverse $\ln ^{-1} \psi z$ is nothing more than $e^{\psi} z$. In this case, Servois observes that the ordinary properties of logarithms follow from his formulas in Section 16.