The following graph shows the number of digits in as a function of .

When I was in school, I stared at this graph for weeks, if not months, trying to figure out an equation that would fit these points. And I never could figure it out.

In retrospect, my biggest mistake was thinking that the formula had to be something like , where the exponent was a little larger than 1. After all, the graph is clearly not a straight line, but it’s also not as curved as a parabola.

What I didn’t know then, but know now, is that there’s a really easy way to determine to determine if a data set exhibits power-law behavior. If , then

.

If we make the substitutions , , and , then this equation becomes

In other words, if the data exhibits power-law behavior, then the log-transformed data would look very much like a straight line. Well, here’s the graph of after applying the transformation:

Ignoring the first couple of pots, the dots show an ever-so-slight concave down pattern, but not enough that would have discouraged me from blindly trying a pattern like . However, because these points do not lie on a straight line and exhibit heteroscedastic behavior, my adolescent self was doomed to failure.

I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.
View all posts by John Quintanilla

Published

One thought on “The number of digits in n! (Part 2)”

## One thought on “The number of digits in n! (Part 2)”