Common logarithms

You know, humans have 10 fingers, generally speaking. And it’s not a coincidence that our number system is based on 10. And for that very same reason, it’s the reason that logarithms to base 10 have historically been the most useful in actual computation. We’ll explain why in a few minutes. So logarithms to base 10, log 10 x, are usually denoted without writing the 10, just log 10. It’s understood that they’re with respect to base 10, and they’re called common logarithms. The primary reason for their great utility is that they convert multiplication to addition and division to subtraction. We’re going to illustrate this. So let’s recall the definition.

To say that y is log x is equivalent to saying that 10 to the y is equal x. Now let’s look at this example. We wish to calculate to a certain accuracy what we get when we divide the number 133352 by 20022. One way to perform this calculation is by the algorithm called long division which you have probably learned in your youth, and it goes a little bit like this. You place the dividend, that’s the larger number here, and the divisor, the other one, next to one another on the paper. You can put it on the left, the divisor, or on the right. It sort of depends on where you grew up, but it doesn’t change the basic algorithm.

And now you do the following. You say, how many multiples of the divisor can I take integral multiples, like 6 times the divisor, and still have a number less than the dividend but not bigger? And it turns out to be 6, after a little estimation. So you take that 6 and you multiply it by the divisor and you get this number. And you draw a line and you subtract. Now, of course, the 13000 something can no longer be divided by the 20000 something. So you do something called bringing down a 0. And the first time you do that tells you where you should place the decimal point in the answer that you are recording above.

And now you continue the process. The 20000 something goes into the 132000 something 6 times. That gives you another 6. You multiply, you subtract, you bring down zeros, sometimes more than one. And after a while you reached a stage in this algorithm where you’ve produced in the answer 6 digits. What you do is you round to 5 digits. And that you call your answer. 6.6603 is the answer to 5 significant digits, as they say. Now you notice this is a fairly long procedure. Well, it is called long division after all.

And it involved 4 multiplications, 6 divisions, 4 subtractions, and also a certain number of steps, at least 4 of them, where you estimate or you introduce and use 0 and that sort of thing. This would be very tedious if you had to do these kinds of divisions all day long. It turns out there’s a much easier way based on the use of logarithms. Let’s see how that would work. Now in order to use this method you need to have a table of logarithms, a log table, so to speak. And it’s actually not a table, it’s a book of logarithms. I have one at home. It’s wonderful reading for falling asleep at night.

So I have a book of logarithms to 5 places of all the integers between 10000 and 100000. Now the logs of these numbers are all between 4 and 5. Why? Because 10 to the fourth is 10000 and 100000 is 10 to the fifth. So you use your book of logarithms and you look up the logarithm of 13335 which is in there. And you get this number 4.12, and so forth. Now that’s not actually the dividend that you’re looking for, the 13000. Your dividend is roughly 10 times bigger. The only difference is the 2 there is in the sixth place. It won’t make a big difference.

And I claim that the logarithm of the dividend then will be approximated by adding 1 to the log you looked up. Why is that? Well, the following argument, the log of 10 times a number is the log of the number plus the log of 10, by the logarithm rule for products. But the common logarithm of 10 is 1 because 10 to the 1 equals 10. Therefore you get 5.12499. Now as for the divisor, the 20022, you can look it up directly in the table and you get its logarithm. And you write it under the other one and you subtract both sides.

And low and behold, using the quotient rule for logarithms you have now found the logarithm of the quotient you’re trying to evaluate. The logarithm of your answer is 0.82348. So how do you find your actual answer? Well, you write the logarithm you’ve found by adding 4 and subtracting 4. When you have the 4.8 business, that’s something that you can look up in the tables in the other direction and see what number it corresponds to, 66601. As for the minus 4, it corresponds to minus the log of 10 to the fourth. And by the quotient rule for logarithms, that puts a 10 to the fourth in the denominator of your answer. So there’s your antilog as it’s called.

And you’ve got your answer, 6.6601, which is correct, by the way, to 4 significant digits. Now you notice how easy that was? What did you have to do? You used the log tables 3 times, twice in one direction and once in the other direction. And you performed 1 subtraction. That’s why it’s so much easier. Now you know this method exploits the fact, as you saw, that the log of 10 times a number is the log of the original number plus 1. And this is why, by the way, on the Richter scale, which is a logarithmic scale for earthquakes, it turns out that an earthquake 10 times as strong is 1 more on the scale.

And this property is essential for being able to use a relatively small table of logarithms. That is, the logarithm is between 10000 and 100000, say. Because by changing the position of the decimal point, you can bring any other number within that range, you see, and then just add or subtract units from its logs. So the fact that it’s base 10 that our number system is based on is crucial here. And that’s why it’s logarithms to base 10 that turn out to be useful. You know, when I was a young man, every mathematical nerd and lots of other people had a device that was considered very high tech at the time. It’s called a slide rule.

What it is is a graduated rule in which numbers are placed according to their logarithms. And then you slide one rule over another in such a way that you wind up physically adding the logarithms of 2 numbers and that corresponds to multiplying the 2 numbers, which you look at in reverse. So it was a nifty way to do multiplications and divisions to about 2 or 3 decimal places. Well, I really like my slide rules but I have to admit, the calculators we have now are pretty wonderful. Have we not come a very long way?