Changing this graph equation so it always passes through (1,1)?

Question

I need a graph of $y=d*ln(cx+1)$ where $d$ is automatically calculated from $c$ so that it always passes through $(1,1)$.

ie. Like this but where $d$ is calculated automatically from $c$:

I tried substituting $x=1$, $y=1$ into the equation and solving for $d$, but that just gives me $d=1/ln(cx+1)$ which of course gives a straight line.

What am I missing on how to do this?

Thanks.

Answer 1

From $$y=d\ln (cx+1)$$

If we let $x=y=1$, then we found the relationship

$$1=d\ln(c+1)$$

or $$d=\frac{1}{\ln(c+1)}$$

Desmos link for the graph.