What's The Discrete Math Derivative Equivalent?

[iScience]

$$ƒ = b^n$$
$$ b,n,I ∈ ℤ $$

Condition: Upon choosing a base value b..

$$ n | b^n ≤ I $$
(n is determined based off the value of b to yield the highest ƒ without going over I)
$$1<b<L , L<<I$$
where I is some large number, and L is also sufficiently large such that we want to avoid going through each base integer via trial and error...

How might I determine the base value that yields a value ƒ that is closest to I?

 

[andrewkirk]

I'm not sure exactly what you are asking.
Are you asking how to, given $L,I\in\mathbb{Z}$ with $0<<L<<I$, find the combination $b,n\in\mathbb{Z}$ that maximizes $b^n$ subject to the constraints
1. $1<b<L$; and
2. $b^n<I$

If so then you could set $Hi=\lfloor\log_2 I\rfloor$ and $Lo=\lceil \log_L I\rceil$, then find
$$b^n=max_{Lo\leq m\leq Hi}\bigg( \lfloor I^\tfrac{1}{m}\rfloor\bigg)^m$$

where $n$ is the value of $m$ that delivers that maximum.
That will give a much smaller search space than $1,...,L$.