That logarithm is the whole reason the law is interesting. If choosing took time in proportion to the number of options, twenty would be ten times worse than two and the advice would just be "have fewer things". Because it's logarithmic, going from two options to four costs about as much as going from four to eight, or eight to sixteen. Each doubling costs a fixed amount. The first few options are expensive; the twentieth is nearly free.
William Hick and Ray Hyman worked this out in 1952, in a lab, with people who had practised the task until the mapping between what they saw and what they pressed was automatic. That last part matters more than anyone quoting the law tends to admit, and we'll come back to it after you've done it yourself.
Controlling for the other law
There's a trap in building this. Lay twenty options out in a list and the ones at the bottom are further away and probably smaller, so you'd be measuring Fitts's Law as much as this one — the times would climb and you'd learn nothing about choosing.
So the options sit on a ring. Every one is exactly the same size and exactly the same distance from the centre, and each round starts by clicking the centre, which puts your pointer in the same place every time. The travel is identical whether there are two options or twenty. The only thing that changes is how many there are.
What the lab left out
If your curve came out closer to a straight line than a logarithm, nothing has gone wrong. Hick's participants knew the whole set of alternatives before the trial began and had practised until responding was reflex. You did not. You had to read the ring, which means you were searching as well as deciding, and search through unfamiliar items tends to cost a fixed amount per item — a straight line, not a curve.
Which is the part worth carrying into a real interface. Hick's Law gets cited to justify trimming menus, and the citation is usually doing more work than the research supports: in a menu nobody has memorised, the cost of the twentieth item is not nearly free. It is about the same as the cost of the second.
What it costs you
Fewer options isn't the lesson. Fewer options at once is. Nothing is saved by deleting a feature people need and burying it somewhere worse. Splitting twenty into four screens of five is a real saving; splitting them into four screens where nobody can guess which is which is not.
Grouping only helps if the groups are obvious. A wrong category costs more than no category, because now the search happens twice — once through your labels and again through the items when the label turned out to be wrong.
A recommended option is free speed. Marking one choice as the usual one lets most people stop deciding immediately, and costs the rest nothing. It is the cheapest performance win in interface design and it is almost always resisted on the grounds of neutrality.
The count that matters is what's on the screen, not what's in the system. Eleven things behind a "More" control cost far less than eleven things laid out at once, even though the system has exactly the same eleven things in it either way.
I have shipped a settings screen with thirty-one switches on one page. Every one of them had a reason to exist and I could have defended all thirty-one individually, which is how it got to thirty-one.