Why is the plot of some functions so similar to the plot of ln(x)?

Question

Using https://www.desmos.com/calculator and my calculus knowledge (the integral power rule $\int x^n dx= (x^n+1)/(n+1)+C$ and the exception $\int x^{-1}dx=ln(x)+C$), I have noticed that functions like $1000(x^{0.001})-1000,\ 1000000(x^{0.000001})-1000000$ etc. have a very similar plot to $\ln(x)$. Is there any justification for why it is like that (apart from the integral rule I've mentioned)? Can logarithms of other bases than $e$ be approximated in a similar way? Thanks!

Answer 1

limit of $x(y^\frac{1}{x}) -x$ as x tends to infinity

$x(y^\frac{1}{x}) -x = \frac{((y^\frac{1}{x}) -1)}{ 1/x}$

use l'hopital rule,

$\frac{-\frac{-log (n)} {x^2} n^\frac{1}{x}}{\frac{-1}{x^2}} $

= $log(n) x^{\frac{1}{x}} = log(n) $ at the limit of x to infinity

Answer 2

$\ln x$ can be defined as $\int_1^x \frac{dt}{t}$. So, if you take the limit of your integrals, $$\lim_{n \rightarrow -1} \int_1^x x^n dx = \lim_{n \rightarrow -1} \frac{x^{n +1}}{n + 1} - \frac{1}{n + 1} = \lim_{n \rightarrow -1} \frac{x^{n+1} - 1}{n + 1}.$$ You can show this limit is equal to $\ln x$ with L'Hopital's rule.