ELF.PAP: Proofs of Algebraic Properties of Exp and Log Functions

Properties	Exponential Form	Logarithmic Form
Sum:Product	$\exp_{base}(a+x) = \exp_{base}(a) \cdot \exp_{base}(x)$ ${base}^{a+x} = {base}^a \cdot {base}^x$	$\log_{base}(b\cdot y) = \log_{base}(b) + \log_{base}(y)$ [$b , y \gt 0$]
Difference:Quotient	$\exp_{base}(x-a) =\frac {\exp_{base}(x)}{\exp_{base}(a)}$ ${base}^{x-a} =\frac {{base}^x}{{base}^a}$	$\log_{base}(\frac yb) = \log_{base}(y) - \log_{base}(b)$ [$b , y \gt 0$]
Product:Power	$\exp_{base}(x\cdot p) =(\exp_{base}(x))^p$ ${base}^{x \cdot p} =({base}^x)^p$	$\log_{base}(y^p) = p \cdot \log_{base}(y) $ [$ y \gt 0$]

The change of basis: $\log_{base}(x)= \frac{\log_c(x)}{\log_c({base})}$.

These examples use the key concept of the inverse relation for the functions $\exp_{base}$ and $\log_{base}$.

Proof ELF.AP.1 Proof of Sum:Product Property of Exp and Log Functions

Proof ELF.AP.2 Proof of Difference:Quotient Property of Exp and Log Functions

Proof ELF.AP.3 Proof of Product:Power Property of Exp and Log Functions

Proof ELF.AP.4 Proof of Change of Basis Property of Log Functions

Exploration of $exp_{base}$ and $log_{base}$ functions through compositions with the core linear functions"$l_{+ a}$ or $l_{\cdot b}$" can also help understand these algebraic properties in a function form.

Linear Function Composition	Exponential Form	Logarithmic Form
Sum:Product	$\exp_{base} (l_{+a}(x)) =\exp_{base}(a+x) = \exp_{base}(a) \cdot \exp_{base}(x) =l_{\cdot \exp_{base}(a)}( \exp_{base}(x)) $	$\log_{base}(l_{\cdot b}(y)) = \log_{base}( b\cdot y) = \log_{base}(b) + \log_{base}(y) = l_{+log_{base}(b)}(\log_{base}(y))$ [$b , y \gt 0$]
Difference:Quotient	$\exp_{base} (l_{-a}(x))= \exp_{base}(x-a) =\frac {\exp_{base}(x)}{\exp_{base}(a)} = l_{\cdot 1/ \exp_{base}(a)}( \exp_{base}(x)) $	$\log_{base}(l_{\cdot 1/b}(y)) =\log_{base}(\frac yb) = log_{base}(y) - log_{base}(b)= l_{-log_{base}(b)}(\log_{base}(y))$ [$b , y \gt 0$]
Product:Power	$\exp_{base}(l_{\cdot p} (x)) =\exp_{base}(x\cdot p)= (\exp_{base}(x))^p =\exp_{\exp_{base}(x)}(p)$	$\log_{base}(\exp_y(p)) = \log_{base}(y^p)= p \cdot \log_{base}(y) =l_{\cdot p}( \log_{base}(y)) $ [$ y \gt 0$]