Function design: a logarithm asymptotic to one? I want to design an increasing monotonic function asymptotic to $1$ when $x\to +\infty $ that uses a logarithm.
Also, the function should have "similar properties" to $\dfrac{x}{1+x}$, i.e.:


*

*increasing monotonic

*$f(x)>0$ when $x>0$

*gets close to 1 "quickly",
    say $f(10)>0.8$


How about
$$
f(x)=\frac{a\log(1+x)}{1+a\log(1+x)},\quad a>0?
$$
Notes:


*

*I have used $\log(1+x)$ instead of $\log x$ to avoid issues near $x=0$ and to make it more similar to $x/(1+x)$.

*Choose $a$ large enough to have $f(10)>0,8$.

*You can see the graph of $f$ (for $a=1$) compared to $x/(1+x)$.



