(related to Proposition: Functional Equation of the Natural Logarithm)
According to the functional equation of the exponential function, we have \[\exp(\xi+\eta)=\exp(\xi)\cdot \exp(\eta)\quad\quad ( * )\] for all real numbers \(\xi,\eta\in\mathbb R\). Because the natural logarithm \[\ln:\mathbb R_{+}^*\to\mathbb R.\] is its inverse function, by setting \[\begin{array}{rcl}x=\exp(\xi)&\Longleftrightarrow&\xi=\ln (x),\\ y=\exp(\eta)&\Longleftrightarrow&\eta=\ln (y), \end{array}\]
With these substitutions and the equation \( ( * ) \), we get: \[\begin{array}{rcl} \exp(\xi+\eta)&=&x\cdot y\\ \ln(\exp(\xi+\eta))&=&\ln(x\cdot y)\\ \xi+\eta&=&\ln(x\cdot y)\\ \ln (x)+\ln (y)&=&\ln(x\cdot y) \end{array}\] for all positive real numbers \(x,y\in\mathbb R_+^*\). It follows that the natural logarithm fulfills the functional equation $\ln(xy)=\ln(x) + \ln(y).$
