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 $f _{log,b}(c)=0$.or

$f_{log,b}(x) = a \cdot log_b(x-h) + k$

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.