Let’s start with a pretty simple function. It’s not this week’s official Sunday Function, but we’ll use it to get there.
Take the number 1, divide it by x. Pretty easy. Now imagine putting a dot at the point x = 1 on the x axis, and draw a line straight up from there to the curve formed by the graph of the function. Then do the same thing for another point on the x axis (with x > 1). Shade in the area in between the lines. Ok, that might be a little hard to visualize if you’ve never done it before. Let me pick the other number to be x = 4, and I’ll actually put the picture below the next paragraph.
Now you or I could have picked any number, not just 4. It stands to reason that the bigger the number we picked, the more area will be in the shaded region. The area of that shaded region is itself a function. We ought to give it a name. How about log(x)?
Log(x), huh? Isn’t there already a function with that name? Yes, the natural logarithm. It’s this very function. Our Sunday Function! Now let’s try to qualitatively determine some of its properties. As x grows larger, our function 1/x gets very small. That means we’re not adding area very rapidly as we extend the shaded region farther to the right. The log function is in fact very slowly growing. It never stops, and if you pick an x big enough log(x) will also get very large. But it does so quite slowly. If you want log(x) = 10, x is going to have to be more than 22,000.
Since this is a natural logarithm, we might expect the number e to be lurking about somewhere. It is. If the total shaded area is 1, the x you will have to have picked is e = 2.718…
Neat, huh? The world of mathematics yields some unusual jewels, and one of my favorites is this relationship between the humble 1/x and the dramatic and ubiquitous exponential function.
[Note: I have followed the standard convention that the natural logarithm is denoted as log(x). In introductory treatments it’s usually denoted ln(x).]