(related to Proposition: Functional Equation of the Exponential Function of General Base)
Let \(x,y,a\) be real numbers and let \(a > 0\). Due to the functional equation of exponential function, we have \[\exp(x+y)=\exp(x)\cdot \exp(y).\] By definition, the exponential equation of general base is given by \[\exp_a(x):=\exp(x\ln(a)).\] From the distributivity law for real numbers, it follows
\[\begin{array}{rcll} \exp_a(x+y)&=&\exp((x+y)\ln(a))&\text{by definition of }\exp_a\\ &=&\exp(x\ln(a)+y\ln(a))&\text{using the distributivity law}\\ &=&\exp(x\ln(a))\cdot\exp(y\ln(a))&\text{applying the functional equation for the exponential function}\\ &=&\exp_a(x)\cdot\exp_a(y)&\text{by definition of }\exp_a\\ \end{array}\]
