One parametric family that interpolates continuously between identity and natural logarithm on (0,1]

Question

I am looking for a family of continuous functions $f_p$, $(0,1]\to\mathbb{R}$, and $p\in [1,\infty)$ that fulfill

$$ f_1 \equiv \log(x) \\ \lim_{p\to \infty} f_p \to x$$

for $x\in (0,1]$. I appreciate useful answers at any level of generality.

Thank you for your consideration.

Answer 1

You could use $f_p(x)=(1/p)\log(x)+(1-1/p)x.$

This is based on affine sums for example $(1-\lambda) A + (\lambda) B$ for $\lambda \in [0,1]$ goes from $A$ to $B.$ Since in your case you wished your parameter $p$ to range in $[1,\infty),$ the idea had to be altered. Note that as $p$ goes in that range, $1/p$ starts at $1$ and goes down toward $0,$ and that the complement $1-1/p$ will go from 0 up toward $1.$