Linear Composition with Core Exponential and Logarithmic
Functions
A general exponential or logarithmic function is a composition of core linear functions with a core exponential or logarithmic
function.
Simple modifications of $exp_b$ or $log_b$ functions can
involve compositions with the core linear functions, giving
functions in a "vertex" form similar to that for quadratic polynomials:[$a \ne 0$]
$f_{exp,b}(x) = a \cdot exp_b(x-h) + k$
or
$f_{log,b}(x) = a \cdot log_b(x-h) + k$
The function $f_{log,b}$ always has a unique root, when $ c= h +
b^{-\frac k a}$ then $f _{log,b}(c)=0$.
When $a \cdot k < 0 $, then $f_{exp,b}$ has a unique root, when $
c = h+ log_b( -\frac k a)$ then $f _{exp,b}(c)=0$
If $k = 0$ then $f_{log,b}$ will have exactly on root: $c = h-1$
while $f_{exp,b}$ will have no roots if $a \cdot k \ge 0$.
We explore how mapping diagrams for linear compositions with core
exponential and logarithmic functions help us visualize and
understand further these
functions.
Example
ELF.DLCELF.0 : Dynamic Visualization Linear Composition with Core Exponential and Logarithmic Functions: Graphs and Mapping Diagrams