How can I figure out what the log function being used based off the X and Y values?

Question

I have a chart where Microsoft .NET has automatically scaled it using a (supposedly) log10 function of some kind. I need to figure out what formula they're using for the value at each tick mark. The scale of the chart looks like this:

https://i.stack.imgur.com/IVjZr.png

I can assign the tick mark number as X, then value shown as Y and that gives us this table of numbers:

X | Y
1 | 100
2 | 126
3 | 158
4 | 200
5 | 251
6 | 316
7 | 398
8 | 501
9 | 631
10 | 794
11 | 1000
12 | 1259
13 | 1585
14 | 1995
15 | 2512
16 | 3162
17 | 3981
18 | 5012

Which makes a nice little curved growth: https://i.stack.imgur.com/CR6ZU.png

Which is a logarithmic curve, though rotated 180 from what I would expect since it starts slowly and has sharp uptick, instead of slowing over time.

So, how can I figure out f(x) = y if I know those numbers above?

Apologies if my verbiage is off, I haven't touched this type of mathematics in a couple of decades.

Answer 1

The growing appears exponential so let's suppose that $$f(x)=a\cdot b^x$$ Solving $f(1)=100,\;f(11)=1000\;$ gives: \begin{align} a\cdot b^1&=100\\ a\cdot b^{11}&=1000\\ \end{align}

Deduce that $\;b^{10}=10\;$ so that $b=\cdots$ and $\;a=\dfrac {100}b$.
Try the function for other values.