I'm trying to write a function that has grows somewhat logarithmically from an initial value to a final value. I know only roughly what the initial and final values should be and how I want the shape function's of the graph to look, but everything I've tried either grows too fast or continues growing well after my final values.
It's been several years since I've done anything beyond basic algebra, derivatives and matrix operations, but (with the help of Google), I remember the basic function transformations ($af(x - h) + v$). However, these do not seem to help much, as I'm still having trouble coming up with a formula such that the output scales appropriately. All of my attempts involved starting with a "basic" function which has a similar shape to what I'd like ($ln(x)$, $sqrt(x)$, $1/x$, etc.), then applying transforms to attempt to make it play nice. When that didn't work, I tried using an exponential to get a function that grows up to (roughly) my maximum ($max * (1 - e^(-x))$). Like the first attempts, this still ended up growing too quickly.
So, I'm at a loss. I don't remember the process I should be using to get the function I'm looking for and it feels like I'm just trying stuff randomly at this point. How do I go about building this thing?
Also, I apologize for being vague with my information. I'd like to provide more details, but there simply aren't any. This is for a game I'm attempting to develop, so the values aren't set in stone, nor is the shape I've suggested. I'll probably end up repeating this process as I find myself needing different values and growth, hence my focus on the process rather than a concrete solution.
Edit: A simple example that might help express my intent: The function in question is to be used for a bonus reward to be given to a player as they build a chain of "kills" or victories. I expect players to easily build chains around 1-10 without much effort, so I want there to be some noticeable growth to show the player that longer chains = higher reward. At around chains of 60-80 is where I'd like to see the reward maximize. The problem (and why I've not stated these values up to now) is that certain victories aren't worth as much as others, to prevent players from beating on weaker players for easy rewards. So their chain is expressed with a point system: That 1-10 chain will be somewhere around 80-120 points, and the 60-80 chains will be around 550-750. In terms of the function/graph, the input/x is the player's current chain in points and the output/y the reward.
Side note: I'm not even remotely sure I'm asking the question properly, nor do I know what tags should be added. If someone knows something that would better summarize what I'm asking, please feel free to correct the title and tags. :(