Equally spacing family along line How do you space the family, $\phi_s(x)=e^{s/lnx}$, for $\phi,x \in \Bbb R(0,1), s\in \Bbb R(0,\infty) $, such that the distance between any function in the family is the same distance apart, along the line $y=x?$
My solution:
I began by solving for $s$ and got $s=(ln\phi)(lnx).$ Then I simplified it to $s=(lnx)^2$ because of symmetry. Then I divided, $x$ into equally spaced segments along $x\in(0,1)$ and plugged each value of $s$ that I got back into $\phi_s(x)=e^{s/lnx} $. This method passes the eye test but I want to make it more rigorous. 
How exactly does this process work? How do you take the limit of this process so that an infinite number of curves are equally spaced apart with infintesimal distance between curves along $y=x$?
Image 1: Note how the gray nodes are equally spaced apart along the red line $y=x.$

Image 2: More functions are added while retaining equal spacing along $y=x$. Notice how the distance between all functions along $y=x$ decreases from Image 1 to Image 2. I think this distance upon successive iterations should tend to infinitesimal distance between all functions along $y=x$.
 A: Your method of solving $s=(\ln(x))^2$ and dividing the interval $(0,1)$ into equal parts is spot on. That process works because your question can be rewritten as "For a given $n$, how do I select the $n$ curves in the family $\phi_s(x)$ where the $k$-th curve passes through $\left(\frac k{n+1}, \frac k{n+1}\right)$?" Note that the last property is the same as "being equally spaced on $y=x$". With the question re-written, it's easy to see that your math finds those curves.
As to your question about a limiting process, I think a natural approach would be to take your favorite definition of "evenly spaced" on the reals and apply the exact same transformation to find your distribution of $s$ values. For example, arguably the rationals in $(0,1)$ are evenly spaced, so you could select the functions corresponding to $s\in\{(\ln(x))^2:x\in\mathbb{Q}\cap(0,1)\}$. Any time you make the transition to the infinite you have choices to make and need to decide what you want preserved as you make the transition.
