So far, we’ve figured out how to mentally estimate the log of any number greater than 1. What about numbers less than 1? I want to explain how that works, including what happens with 0 or with negative numbers.
First let’s review so far: the logarithm “counts the zeroes” in a number. For a multiple of ten, that is literally true:
log(10) = 1
log(100) = 2
log(1,000) = 3
log(10,000) = 4
log(100,000) = 5
log(1,000,000) = 6
etc.
For numbers that aren’t multiples of ten, we have extended the idea of “number of zeroes”, by using the fact that multiplying numbers will add the number of zeroes: for any two numbers a and b,
log(a x b) = log a + log b.
In words: the “number of zeroes” in a x b is the “number of zeroes” in a plus the “number of zeroes” in b.
For example, the number 3.16 roughly represents half a zero, because
3.16 x 3.16 = 10
approximately. (We are often just rounding this to 3.) Written using the fundamental rule above:
log(3.16 x 3.16) = log (10)
log(3.16 x 3.16) = 1
log(3.16) + log(3.16) = 1 (using the fundamental rule)
so log(3.16) = 1/2 (approximately).
I asked you to find log(1) in a problem above. In case you ran into trouble, here’s how the solution goes:
1 x 10 = 10
so log(1 x 10) = log (10)
so log(1) + log(10) = log(10)
so log(1) + 1 = 1
so log(1) = 0 (after subtracting 1 from both sides).
Therefore log 1 = 0. The number 1 counts as having “no zeroes”. Multiplying 1 by a number doesn’t change it, so it doesn’t add anything to the number of zeroes.
(Note that I could have used any positive number in place of 10 in the argument above; the only reason I used 10 is because I know log(10), but in fact I didn’t really need it: I could have subtracted a log(10) from both sides one line earlier than I did.)
The logs of positive numbers which are less than 1
OK, so what about a number less than 1? Since we’re going with multiples of 10, let’s try 0.1 first. I need to relate 0.1 to something I already know the log of. And I should use multiplication, because I want to use the fundamental rule. So, I’m going to start with
0.1 x 100 = 10.
(That isn’t the only thing I could have started with. In what follows, try something else in place of 100, and see if it works out!)
So now, following the same sort of strategy I’ve been doing so far:
log(0.1 x 100) = log(10)
log(0.1) + 2 = 1
so log(0.1) = -1.
(I skipped steps this time; can you fill them in?)
So, 0.1 counts as “1 negative zero”. Or an “antizero”. You can see that more informally from
0.1 x 100 = 10;
multiplying adds the zeroes. So I added some number of zeroes to 2 zeroes, and got 1 zero as a result. I must have added -1 zeroes.
Problem 1: Find the following. (Even if you have a reasonable guess for what these are, please write out the reasoning. You can either reason like I did for log(0.1); or you can use log(0.1)=-1, and use the fundamental rule.)
a. log(0.01)
b. log(0.001)
c. log(0.0001)
d. log(0.00001)
Problem 2: Find approximate values for the following, without a calculator. Once you find them, check your answers with a calculator.
a. log(0.7)
b. log(0.5)
c. log(0.0035)
d. log(0.00008)
Hopefully you can see that you can now mentally approximately calculate the log of any positive number, no matter how big or how small.
The log of zero??
What you have figured out above should tell you that something weird is going on with the log of zero.
Problem 3:
a. The logs you found in Problem 1 of this lecture are logs of numbers that are getting closer and closer to zero (0.1, 0.01, 0.001, 0.0001, etc.). What does this suggest about the value of log(0)? Why should that make you worried?
b. Remember the graph you did of y=log x? The one where your x range went from 0 to 10, but you could only fill it in starting at 1? Take the values you found in Problem 1, plot the corresponding points on the graph of y = log x, and use them to extend your graph. What is that suggesting about log(0)?
Problem 4: Undeterred, we might try to figure out log(0) directly. We could use a similar strategy to what we tried above: we could start with an equation like, say, 0 x 10 = 0, and follow similar steps to try to find log(0). What happens when you carry through this strategy? You should get a troubling result: what do you think this troubling result means??
The log of negative numbers
If you remember logs from high school, you might recall that it was always forbidden to take the log of a negative number. There are good reasons for this in the high school approach. However, whether logs of negatives makes sense depends a bit on your definitions, and on what you want to use logs for. I’ll come back to this point later. But for the moment, I want to note that, according to our reasoning in this first version of logs, the logs of negative numbers are actually totally fine.
Problem 5: Use the fact that (-1) x (-1) = 1, and the fundamental rule for logs, to find the value of log (-1).
Problem 6: Use the value you found for log (-1), plus the fundamental rule, to find:
a. log (-10)
b. log (-100)
c. log (-1,000)
Problem 7: Based on what you were just doing, you hopefully saw a simple rule. If x is any positive number, state the rule for what log(-x) is. (You should be able to state it as a formula.) Explain why your rule is always correct (your explanation should take the form of some equations).
