In physics and economics courses, we often calculate the "percentage difference" or "relative change" of two numbers. Different teachers instruct students to do it in different ways. If we want to calculate the percentage increase of a variable $x$ when its value increases from $x_0$ to $x_1$($x_1>x_0$), the simplest way to do so is $$ \mu=\frac{x_1-x_0}{x_0}(\times100\%). $$ But often "midpoint formula" is preferred: $$ \lambda=\frac{x_1-x_0}{\frac{x_0+x_1}{2}} $$ We can show that $$ \lambda=\frac{\mu}{1+\mu/2}. $$ However, I think we can also use geometric mean instead of the arithmetic mean in the denominator of $\lambda$. So let's define $$ \nu=\frac{x_1-x_0}{(x_0x_1)^{1/2}}=\frac{\mu}{(1+\mu)^{1/2}}. $$ However, in my opinion, the best way to calculate the total percentage change is to add the percentage change of every small increment of $x$ together. The percentage change when $x$ increases to $x+dx$ is $dx/x$, so the total percentage change of an increase from $x_0$ to $x_1$ is $$ \xi=\int_{x_0}^{x_1}\frac{dx}{x}=\ln(\frac{x_1}{x_0})=\ln(1+\mu) $$ $\mu$, $\lambda$, and $\nu$ are all approximation of $\xi$.
Since $$ \mu=\mu\\ \lambda=\mu-\frac{\mu^2}{2}+\frac{\mu^3}{4}-...\\ \nu=\mu-\frac{\mu^2}{2}+\frac{3\mu^3}{8}-...\\ \xi=\mu-\frac{\mu^2}{2}+\frac{\mu^3}{3}-... $$ $\lambda$ and $\nu$ are better approximation of $\xi$ than $\mu$. However it appears to me that $\nu$ is also better than $\lambda$ as an approximation.
Question: Which of the quantities above is the best measure of percentage increase? Is there a branch of math that studies this?