Algebraic Properties of Exponential and Logarithmic Functions: Proofs
 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: $\log_{base}(x)= \frac{\log_c(x)}{\log_c({base})}$.

Visualizations of the arguments used to justify these algebraic identities using mapping diagrams are presented in the following examples.

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 $\exp_{base}(l_{\cdot p} (x)) =\exp_{base}(x\cdot p)= (\exp_{base}(x))^p =\exp_{\exp_{base}(x)}(p)$ $\log_{base}(\exp_y(p)) = \log_{base}(y^p)= p \cdot \log_{base}(y) =l_{\cdot p}( \log_{base}(y))$ [$y \gt 0$]