Algebraic Properties of Exponential and Logarithmic Functions

A core exponential or logarithmic function has some important algebraic properties with regard to addition and multiplication which are commonly treated whenever the logarithm is introduced.
These key properties played a very significant role as a tool for simplifying calculations since the early 17th century.


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$]

Another key property of logarithmic functions is the change of basis result that $\log_{base}(x)= \frac{\log_c(x)}{\log_c({base})}$.

We explore how mapping diagrams for core linear functions composed with core exponential and logarithmic functions help us visualize and understand further these algebraic properties.
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$]

Example ELF.AP.1 : $\exp_3(2+x) = \exp_3(2) \cdot \exp_3(x) = 3^{2+x} = 3^2 \cdot 3^x$ and $\log_3(9y) = \log_3(9) + \log_3(y)$

Example ELF.AP.2 : $\exp_3(x-2) =\frac {\exp_3(x)}{\exp_3(2)} = 3^{x-2} =\frac {3^x}{3^2}$ and $\log_3(\frac y9) = \log_3(y) - \log_3(9)$

Example ELF.AP.3 : $\exp_3(x \cdot 2) =(\exp_3(x))^2 = 3^{x \cdot 2} =(3^x)^2$ and  $\log_3(y^2) = 2\cdot \log_3(y) $
Mapping diagrams can also help understand the proofs of the logarithmic identities as the following proofs demonstrate.
These examples use the key concept of the inverse relation for the functions $\exp_{base}$ and $\log_{base}$.

Proof ELF.AP.P.1 : Sum:Product Property of Exp and Log Functions  $\log_{base}( y\cdot b ) = \log_{base}(y) + \log_{base}(b)$

Proof ELF.AP.P.2 : Difference:Quotient Property of Exp and Log Functions  $\log_{base}( \frac yb ) = \log_{base}(y) - \log_{base}(b)$

Proof ELF.AP.P.3 : Product:Power Property of Exp and Log Functions $\log_{base}( y^p ) = p\log_{base}(y) $

Proof ELF.AP.P.4 : Change of Basis Property $ \ln(y) = \log_{base}( y ) \cdot ln(base) $ or $\log_{base}(y) = \frac{\ln(y)} {\ln(base)}$