# Unifying Aspects of Generalized Calculus
## 3 Non-Newtonian Exponential Function and Logarithm  
Once we know how to differentiate and integrate, we can turn to differential equations. The so-called exponential family plays a crucial role in thermodynamics, both standard and generalized [39; 40; 41; 42; 43]. Many different deformations of the usual \(e^{x}\) can be found in the literature. However, from the non-Newtonian perspective, the exponential function \(\mathrm{Exp}:\mathbb{X}\to\mathbb{Y}\) is defined by  
\[\frac{\mathrm{D}\mathrm{Exp}(x)}{\mathrm{D}x}=\mathrm{Exp}(x),\quad\mathrm{ Exp}(0_{\mathbb{X}})=1_{\mathbb{Y}}. \tag{25}\]  
Integrating (25) (in a non-Newtonian way) one finds the unique solution  
\[\mathrm{Exp}(x)=f_{\mathbb{Y}}^{-1}\left(e^{f_{\mathbb{X}}(x)}\right),\quad \mathrm{Exp}(x_{1}\oplus_{\mathbb{X}}x_{2})=\mathrm{Exp}(x_{1})\odot_{\mathbb{ Y}}\mathrm{Exp}(x_{2}). \tag{26}\]  
In thermodynamic applications, one often encounters exponents of negative arguments, \(e^{-x}\). In a non-Newtonian context the correct form of a minus is \(\ominus_{\mathbb{X}}x=0_{\mathbb{X}}\ominus_{\mathbb{X}}x=f_{\mathbb{X}}^{-1} \big{(}-f_{\mathbb{X}}(x)\big{)}\). The example discussed in the next section will involve \(\mathbb{X}=\mathbb{R}\) and \(f_{\mathbb{X}}^{-1}(-r)=-f_{\mathbb{X}}^{-1}(\mathbf{r})\). In consequence, it will be correct to write \(\ominus_{\mathbb{X}}x=-x\), but in general such a simple rule may be meaningless (because \"\(-\)\", as opposed to \(\ominus_{\mathbb{X}}\), may be undefined in \(\mathbb{X}\)). **Example 4**. : _Let \(\mathbb{X}=(\mathbb{R}_{+},\oplus,\odot)\), with the arithmetic defined by \(f_{\mathbb{X}}:\mathbb{R}_{+}\to\mathbb{R}\), \(f_{\mathbb{X}}(x)=\ln x\), \(f_{\mathbb{X}}^{-1}(r)=e^{r}\). Then_  
\[\ominus_{\mathbb{X}}x=f_{\mathbb{X}}^{-1}\big{(}-f_{\mathbb{X}}(x)\big{)}=e^{ -\ln x}=1/x\in\mathbb{R}_{+}. \tag{27}\]  
_The same number can be both positive and negative, depending on the arithmetic._A (natural) logarithm is the inverse of \(\mathrm{Exp}\), namely, \(\mathrm{Ln}:\mathbb{Y}\to\mathbb{X}\),  
\[\mathrm{Ln}(y)=f_{\mathbb{X}}^{-1}\left(\ln f_{\mathbb{Y}}(x)\right), \quad\mathrm{Ln}(y_{1}\odot_{\mathbb{Y}}y_{2})=\mathrm{Ln}(y_{1})\oplus_{ \mathbb{X}}\mathrm{Ln}(y_{2}). \tag{28}\]  
Expressions such as \(\mathrm{Exp}\ x+\mathrm{Ln}\ y\) are in general meaningless even if \(\mathbb{X}\subset\mathbb{R}_{+}\) and \(\mathbb{Y}\subset\mathbb{R}_{+}\). However, formulas such as  
\[\left(\mathrm{Exp}\ x\right)\oplus_{\mathbb{Z}}^{\mathbb{Y}\mathbb{X}}\left( \mathrm{Ln}\ y\right)=f_{\mathbb{Z}}^{-1}\left(e^{f_{\mathbb{X}}(x)}+\ln f_{ \mathbb{Y}}(y)\right) \tag{29}\]  
make perfect sense. For example, if \(p_{k}\in\mathbb{X}\), then an entropy can be defined as  
\[S = \bigoplus_{k}p_{k}\odot_{\mathbb{Z}}^{\mathbb{Y}\mathbb{Y}} \mathrm{Ln}\left(1_{\mathbb{X}}\oslash_{\mathbb{X}}p_{k}\right) \tag{30}\] \[= f_{\mathbb{Z}}^{-1}\left[\sum_{k}f_{\mathbb{Z}}\left(p_{k}\odot_ {\mathbb{Z}}^{\mathbb{Y}\mathbb{Y}}\mathrm{Ln}\big{(}1_{\mathbb{X}}\oslash_{ \mathbb{X}}p_{k}\big{)}\right)\right]\] (31) \[= f_{\mathbb{Z}}^{-1}\left[\sum_{k}f_{\mathbb{X}}(p_{k})\ln\left(1/ f_{\mathbb{X}}(p_{k})\right)\right]. \tag{32}\]  
Many intriguing questions occur if one asks about normalization of probabilities. We will come to it later. Non-Newtonian constructions of \(\mathrm{Exp}\) and \(\mathrm{Ln}\) are systematic, general, and flexible. There seems to exist a relation between the arithmetic formalism and the method of monotone embedding discussed in information geometry [44], but the problem requires further studies. **Example 5**. : _In order to appreciate the difference between Newtonian and non-Newtonian differentiation let us differentiate the function \(A(x)=x\), \(A:\mathbb{X}\to\mathbb{Y}\), but in two cases. The first one is trivial, \(\mathbb{X}=\mathbb{Y}=\left(\mathbb{R},+,\cdot\,\right)\), with the arithmetic defined by the identity \(f_{\mathbb{X}}=f_{\mathbb{Y}}=\mathrm{id}_{\mathbb{R}}\).