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.LCELF.1 : $2\cdot \exp_3(x-1) - 4$ and $2 \cdot \log_3(x-1) - 4$.

Example ELF.LCELF.2 : $2\cdot \exp_3(x-1) + 4$ and $2 \cdot \log_3(x-1) + 4$.

Example ELF.LCELF.3 :  $2\cdot \exp_3(x-1)$ and $2 \cdot \log_3(x-1)$.

Example ELF.DLCELF.0 : Dynamic Visualization Linear Composition with Core Exponential and Logarithmic Functions: Graphs and Mapping Diagrams