I need to find a function $f(x)$ such that the following data points would fit on it: $$f(1) = 0 \\ f(2) = 0.5 \\ f(4) = 1.0 \\ f(8) = 1.5 \\ \cdots $$ and so on. So the pattern is every time $x$ doubles $f(x)$ increases by $0.5$.
What is the equation for $f(x)$?
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If you plot $f(x)$ (W|A), the graph should give you a clue that this is a logarithmic function. Also, you can consider the fact that what you have is a recurrence relation: $f(x) = f(x/2) + 0.5$, which you solve using a technique such as Master theorem. Or for example using W|Alpha.
A function where multiplying the argument by a fixed factor results in adding a fixed constant to the output is (more or less by definition) a logarithmic function, not a quadratic function. Specifically, it must be a function of the form $f(x) = c \log x + d$ for some real numbers $c$ and $d$, which you can determine by plugging in a couple of given values.