Exponential increase - equation

Question

I'm smashing my head from hours trying to achieve an exponential increase. I tried with second degree equation but the result is not what i expected. Let me explain.
I have a pay per use service based on credit balance. Users can upload funds anytime to their accounts. I want to incentivize clients to upload more funds to avoid micro-payments and higher transaction fees. To make it short the more you pay, the more bonus you get. For example an user could deposit 100$ in his account in two ways:
Jack: 10 transactions 10$ each
Paul: 2 transactions 50$ each
Mark: 1 transaction of 100$
Paul should get an higher bonus than Jack but less compared to Mark. Now I'm trying with this function.
for($amount=10;$amount<=5000;$amount +=10)
{
$rate=0.005;
$bonus=exp($rate*$amount);
echo "Deposit ".$amount."$ and gain ".$bonus." bonus<br>";
}
Minimum deposit is 10$ and Maximum deposit is 5000$ that's why i loop from 10 to 5000. The problem is simple.
Deposit 10$ and gain 1.05127109638 bonus
...
Deposit 100$ and gain 1.6487212707 bonus
...
Deposit 1000$ and gain 148.413159103 bonus
...
Deposit 2500$ and gain 268337.286521 bonus
...
Deposit 5000$ and gain 72004899337.4 bonus
You get a too little for small amounts and way too much for big amounts. I've also tried with different ranges for example 10 to 100$ with a certain rate, 200 to 1000 with another (...) but of course when you deposit an amount that is near to these limits you get less bonus. It's not logical.
Deposit 1000$ and gain 54.5981500331 bonus
... here starts the next range ...
Deposit 1250$ and gain 42.5210820001 bonus
I've also tried this approach:
function foo($wanted=1000, $rangeLow=10, $rangeHigh=5000){
$increment=($rangeHigh-$rangeLow)/($wanted+1);
$r=array();
for ($i=$rangeLow+$increment;$i<$rangeHigh;$i+=$increment)
$r[]=$i;
return $r;
}
I "spread" 1000 bonus points between 10$ and 5000$ but with this I get a linear increase while I need an exponential one. Probably the solution is mixing both approaches but I don't know how to.

Answer 1

I think you want the pow function instead of exp
pow Returns base raised to the power of exp.
exp Returns e raised to the power of arg.
e.g.
for($amount=10;$amount<=5000;$amount +=10)
{
$rate=0.05; //adjust as needed, 0.08 makes 5000 -> about x2
$bonus=pow($amount, $rate);
echo "Deposit ".$amount."$ and gain ".$bonus." bonus<br>\n";
}
Outputs:
Deposit 10$ and gain 1.122018454302 bonus
Deposit 20$ and gain 1.1615863496415 bonus
Deposit 30$ and gain 1.1853758165593 bonus
..
Deposit 1000$ and gain 1.4125375446228 bonus
..
Deposit 5000$ and gain 1.5309059120639 bonus

Answer 2

Don't you just need to keep your exp function but multiply it by a value less than 1? If I've got my maths right, that should flatten the exponential curve.

Answer 3

I finally managed to complete this task. It's not something that you can achieve only using PHP. You also need an equation otherwise you have no controls over the increases.
In my case i have an equation of parabola with 3 points given where have to define the lowest and the highest value. I wanted to give a maximum bonus of 1000 and a minimum of 0.1 for imports from 10 to 5000$. I did it on Excel with matrix product of a matrix inversion of values. Anyway I'm not going to share the whole formula since it's too specific. I prefer to share with you the PHP-side of the story.
I wanted to "convert" my Excel formula in PHP but it took me too much time to code matrix calculations then i decided look for an existing library. Pear has a Math_Matrix package. If you don't want to recompile your webserver to get Pear, you can probably use this PHP class. I tested matrix inversions and it works.
Said that i suggest you not to use any of the above solutions unless you use matrix math frequently. It's more convenient and efficient to you calculate all necessary values on Excel or with the "old" pen and paper so that you can pass to PHP simple calculations like +, -, *, ^2 etc.