Mokko Theory Episode 192 Act 3: The Abyss of Measure

Scene 1: The Ghost of Stirling and the Third Square Root
On the calculation paper spread across the desk, a new history is being carved, one where the multiplicative world has been converted into an additive one via logarithms.
Let the discrete log history be $${L_d(N)}$$.
$${L_d(N) = \sum_{k=1}^{N}\log k = \log(N!)}$$
Let the continuous log history be $${L_c(N)}$$.
$${L_c(N) = \int_1^N\log x\,dx = N\log N-N+1}$$
Define the difference between these two.
$${R_{\log}(N) = L_d(N) - L_c(N)}$$
Substitute Stirling's formula here.
$${\log(N!) \approx N\log N - N + \frac{1}{2}\log(2\pi N)}$$
The large terms $${N\log N}$$ and $${N}$$ cancel each other out, and the protagonist of the residual reveals itself.
$${R_{\log}(N) \approx \frac{1}{2}\log N}$$
The $${1/2}$$ that appeared naturally not from an averaging operation at the endpoints, but as the difference between the discrete and the continuous.
Let's take the exponent of this residual in the logarithmic world and return to the original world.
$${e^{\frac{1}{2}\log N} = N^{1/2} = \sqrt{N}}$$
The center of the divisor structure, the midpoint of the log history, and the Stirling residual.
Three different routes have all converged to the same destination.
$${\sqrt{N}}$$
Why is this quantity repeated so many times?

Scene 2: One-Lung Flight in the Log World and Doubt Toward Measure
As I stare at the list of drawn formulas, a fundamental distortion becomes apparent.
Even though I claimed to have moved to the log world, the variables being calculated remained $${1, 2, 3, \dots, N}$$. Only the values were changed to logs, while the ruler of the coordinates continued to advance in the additive world.
A "single step" in the additive world is as follows:
$${N \rightarrow N+1}$$
However, a "single step" in the true log world should involve a change in the measure itself.
$${\log(N+1) - \log N}$$
Yet, in integration, I continued to use this infinitesimal quantity without any doubt.
$${dx}$$
If the world were completely logarithmic, wouldn't the natural infinitesimal quantity that should be there be this?
$${d(\log x)}$$
The object of the inquiry is no longer $${\sqrt{N}}$$ or $${1/2}$$.
It is a doubt toward the measure itself: what constitutes a "single step" in each world, and what is the standard for measuring distance?
The problem was not in $${x}$$, but in $${dx}$$.