After posting yesterday’s blog on using logarithms for calculations, I realized I left out a significant wrinkle, one that used to catch me time and again. That’s the logarithms of small numbers. It’s really simple, but it can be confusing. That’s because base 10 logarithms less than 1 are always negative. Always.

To understand why, first a quick refresher of what’s going on with the characteristic and mantissa. Since base 10 logarithms are based on 10, the characteristic is identical to the number multiplied by 10 raised to the power of the characteristic. It’s just like using scientific notation: 200 = 2 x 10²; 20 = 2 x 10¹; 2 = 2 x 10⁰; 0.2 = 2 x 10^-1; and 0.02 = 2 x 10^-2. Since any number raised to the 0 power = 1, then 10⁰= 1.

When we convert these to logarithms, what we raise 10 to becomes the characteristic. The logarithm of 2, to seven significant digits, is 0.301 030 0. So the logarithm of 200 = 2.301 030 0 and of 20 = 1.301 030 0. What, though, is the logarithm of 0.2?

Here is the gotcha. If we write -1.301 030 0, we have the wrong answer. To understand why, remember that multiplying numbers is the same as adding their logarithms. So the logarithm of 200 is the logarithm of 2 x 10^2. This means that when we find the logarithm, it’s actually 0.301 030 0 + 2, since the logarithm of 2 is 0.301 030 0 and the logarithm of 10² is 2. 0.301 030 0 + 2 = 2.301 030 0.

This also means that the logarithm of 0.2 = the logarithm of 2 x 10^-1 = 0.301 030 0 + -1. The logarithm of 0.2 is then -0.698 970 0. And, when I got in a hurry, this would catch me every time.

After we get into negative territory, the mantissa doesn’t change. The logarithm of 0.02 is -1.698 970 0, and the logarithm of 0.002 is -2.698 970 0.

The one consolation is that negative logarithms can often be a point of confusion. Does the logarithms 10.000 000 0 – 0.698 970 0 mean 100 ÷ 5 or 100 x 0.2? Yes, they evaluate to exactly the same answer, but in some cases clarification is needed. For this reason one convention is to note a negative characteristic with a bar across the top. Under this notation, the logarithm of 0.2 becomes 1̄ + 0.301 030 0. Another convention, for characteristics between 0 and  -10, is to add 10 and note this by also subtracting 10 in the equation. By this notation, the logarithm of 0.2 would become 10 + (-1 + .301 030 0) -10 = 10 -1 + 0.301 030 0 – 10. = 9.301 030 0 – 10.

Honestly? I don’t recall seeing either used. Remember, though, I was right on the end of the manual calculation era and at the beginning of the age of pocket calculators. For calculation purposes, you can simply say the logarithm of 0.2 is -1 + 0.301 030 0 = -0.698 970 0, and be done with it.

This brings up the question of whether there’s such a thing as a logarithm of a negative number. That’s impossible for any positive base number. The reason is simple. Remember the rules of signs and multiplying numbers: If both numbers are positive, the results are positive; if both are negative, the results are positive; and if one is positive and the other negative, the result is negative. Since a logarithm is simply the question of what power you need to raise the base to equal a number, if you have a positive base number it becomes impossible to find the logarithm of a negative number. This is also why it’s impossible to take the square root of a negative number.

Actually, that’s not quite right. There’s something called an imaginary number that plays with the idea of what if there was a number that had a negative square root. This is denoted by i and you see it in places like √2i, which essentially means √(2 x -1). This gets into something else, and why your scientific calculator will give you an error if you try to take the logarithm or square root of a negative number.

What do we do, then, if we’re using logarithms for calculations and need to multiply or divide by a negative number? Using logarithms, how do we evaluate 123÷-463? Simple; we cheat a bit. The rules for signs and division are the same as multiplication: If both numbers are positive, the results are positive; if both are negative, the results are positive; and if one is positive and the other negative, the result is negative. Knowing this, we can write 123÷-456 as -1(123÷456), then evaluate as -1(log(123) -log(456) = -1(10(^{2.089 905 1 – 2.658 964 8})) = -1(10^{–0.569 0590 7}) = -1(0.269 736 9) = -0.269 736 9.

All good things to keep in mind when using logarithms. After all, if we must be negative, we’d best do it right.