[Math] Connection between linear/quadratic/cubic/logarithmic convergence and function

Question

1. From the way linear/quadratic/cubic convergence of a sequence are
   defined, I wonder why they are called linear/quadratic/cubic, in the
   sense of some connections to linear/quadratic/cubic functions.

   Here are the definitions of linear/quadratic/cubic convergence of a
   sequence in my words based on Wikipedia

   Suppose that the sequence $\{x_k\}$ converges to the number $L$.
   Suppose $q > 1$.

   When $\lim_{k\to \infty} \frac{|x_{k+1}-L|}{|x_k-L|^q} = μ$ and $μ ∈ (0, 1)$, we say that the sequence (Q-)converges linearly if
   $q=1$,
   quadratically if $q=2$, and cubically if $q=3$.

2. Similarly, how is logarithmic convergence connected to a logarithm
   function? The definition of logarithmic convergence is from the same
   link to Wikipedia:

   If the sequences converges sublinearly and additionally $
   \lim_{k\to \infty} \frac{|x_{k+2} – x_{k+1}|}{|x_{k+1} – x_k|} = 1, $ then it is said the sequence $\{x_k\}$ converges logarithmically
   to $L$.

I found a plot of linear, linear, quadratic and logarithmic rates of convergence for an example in Wikipedia, which seems to
suggest some connection, although it is not clear to me how they are
connected:

Thanks for clarification!

Answer 1

It's simply that $x^q$ is a linear function of $x$ if $q=1$, a quadratic function of $x$ if $q=2$ and a cubic function of $x$ if $q=3$. Convergence "logarithmically" is different, and that definition really doesn't have any necessary connection to a logarithm function.