Not to blame him, but I promised Peter Nena that I’d tell my favorite story that involves logarithms.
Hang on. Don’t skip to the gallery just yet., Give me a chance to make this interesting.
Do you remember what logarithms are? If so, skip to the story. If not, you’re making it harder for me to keep this interesting. I’m going to give you a break. I am going to explain what logarithms are but I’m not going to explain why they’re good things to have and how they make our lives easier. I am only going to explain the little bit about logarithms that you need to know to appreciate my story. OK? Is it a deal? And, to make life even better, I’m going to only address logarithms in base 10, you know, the normal number system we live with.
Logarithm definition – a quantity representing the power to which 10 must be raised to produce a given number.
Example: the logarithm (log) of 100 is 2. 102 is 100. The log of 1,000 is 3. 103 is 1,000. See the pattern?
People are generally comfortable up to this point. The problem comes when you want to know the log of 529. That’s when we have to accept that we can raise 10 to a power that’s between 2 and 3. It’s 2.7234. Yep, 102.7234 is 529. I see your eyes rolling. I sense you trying to grasp multiplying 10 by itself 2.7234 times. I think I just felt half of you skip to the gallery. If you’re still here, hang one for one more thing about logarithms – then I’ll be done, I promise.
Doing simple math with logarithms is easy and powerful. To multiply two numbers, you add their logarithms. To divide one number into another, you subtract Log-a from Log-b. Take my word for it. Here’s a simple example.
12 x 44 = ? (hint, it equals 528)
The log of 12 = 1.079
The log of 44 = 1.643
If we add those, we get 2.722
Now, raise 10 to the 2.722 power (102.722) = 527.2298 Note: it would be 528 if I used more numbers to the right of the decimal point in those log values.
That’s it – that’s the end of math. Now we get onto the story which, unfortunately involves more math and a little science.
When I was in college, I was a chemistry major. I was working toward a chemistry degree, concentrating in “computer assisted analytic chemistry.” This was 1974 and my professor and I were making things up as we went along. He gave me a research project that had been sponsored by an industry group. The project had been assigned to a graduate student who did some work, but never finished. The goal was to write a computer simulation of a chemical process called Countercurrent Distribution. I won’t go into the process other than to say that it’s way of separating liquids from other liquids, it helped in the development of Penicillin and it’s kind of how your kidneys work.
In any case, it’s a long tedious process involving, at the time, an elaborate apparatus. The industry that was paying for our research typically used a machine capable of 1,000, oh, let’s call them cycles. The apparatus in our lab could manage 250 cycles, but we were trying to predict where among the 1,000 places the liquid we were looking for would settle. It was hard to extrapolate the results from 250 cycles up to 1,000. Worse, running 250 cycles took 40 hours. Hence the desire to simulate the apparatus.
The industry was using the Standard Bell curve (remember that) to approximate the results. Unfortunately, the real results were skewed off to one side of the center and the peak of the bell was higher and narrower. My job was to predict where the peak would be, and how high and wide it would be. This was easy, except for one thing – well, two things.
At each end of the Standard Bell curve, and my simulation, the number along the X-axis approaches infinity (and negative infinity) while the number on the Y-axis approaches zero. In my simulation, the number on the Y-axis had to be divided into some other number, and computers give up when asked to divide by zero, or something so close to zero that it looks like zero.
This problem is why the graduate student never finished the project.
I spent from September to November trying to work around this problem. Finally, it occurred to me to change all the math to use logarithms. Since dividing while using logarithms involves subtraction, and computers were quite capable of subtracting zero from anything, it worked.
During my review with my professor in December, he asked if I had made any progress. I sheepishly offered, “I got the program working, but I haven’t had much time to play with the process.” I was hoping the progress I had made would be good enough for a B.
He got excited. “You got it working? How did you do that?” I explained the use of logarithms and he did the 1974 equivalent of a head slap. He reached over, shook my hand, and said, “you just earned four A’s!” Then he explained that being able to tell the industry group we had a working model would release the next wave of funding. I spent three more semesters adding features to my simulation. I received an A each semester.