This page contains introductory material. If you are looking for more advanced math, please see the Advanced Logarithms page.
Have you ever heard someone say that they hope to make a six-figure salary someday? What they are hoping for is to make $100,000 or more. The reason this is called a six-figure salary is because if you count the digits in the number, you need six digits to write it down: a one and five zeros. Pretty simple. Making seven figures would be even better, since that would mean at least a million per year in your pocket.
But this term is not very precise; making $350,000 per year, for example, would be much better than $100,000, but that would also be considered a six-figure salary. Even $999,999 per year is technically six figures, though it is just a dollar shy of being a seven-figure salary.
A similar concept is to count how many zeros are in numbers like 100 or 1000 (numbers that start with a one and are followed by zeros). Here is a table of such numbers along with a count of their number of zeros:
| Number | Zeros | |
|---|---|---|
| 10 | 1 | |
| 100 | 2 | |
| 1,000 | 3 | |
| 10,000 | 4 | |
| 100,000 | 5 | |
| 1,000,000 | 6 |
Counting the zeros in a number is obviously a very simple thing to do, but how could something so simple actually be useful? It turns out that this simple concept is one of the most powerful in all of mathematics. In the paragraphs that follow, we'll see how paying attention to the count of zeros in numbers reveals hidden relationships that allow us to solve seemingly difficult problems with ease.
Let's see what happens when we multiply two of these numbers together. For example, if we multiply 100 times 10, we get 1000:
| Number | Zeros | ||
|---|---|---|---|
| 100 | 2 | ||
| × | 10 | 1 | |
| 1,000 | 3 | ||
Did you notice what happened to the number of zeros? When we multiplied the two numbers together, the number of zeros in the result is just the sum of the number of zeros in each of the original numbers (2 + 1 = 3). You already knew that multiplying a number by 10 just adds a zero to the end of the number, and that multiplying by 100 adds two zeros, so it makes sense that this works in general.
Another thing to notice about these numbers is that they are powers of 10. This means that you can calculate each number by repeatedly multiplying by ten. For example, one thousand is:
1000 = 10 × 10 × 10
Notice that the number of zeros in 1000 is the same as the number of 10's that are multiplied together, in this case 3.
It is tedious to have to write 10 × 10 × 10 and also wastes space, so a long time ago someone really smart came up with the idea of the exponent to indicate the number of times to multiply a number together. An exponent follows the number and is placed above the midpoint like so: 1000 = 103. An exponent is also called a power. You would say, "1000 is 10 to the third power."
Here again is our table of numbers, but now we've added some columns to show the multiples of 10 and the equivalent exponents:
| Number | Multiples of 10 | Power | ||
|---|---|---|---|---|
| 10 | = | 10 | = | 101 |
| 100 | = | 10 × 10 | = | 102 |
| 1,000 | = | 10 × 10 × 10 | = | 103 |
| 10,000 | = | 10 × 10 × 10 × 10 | = | 104 |
Notice that the power for each number is just the number of zeros. Many calculators will have a button to compute powers of 10 for you. It will most likely be labeled 10x. First you enter a number for the power (exponent), then press the 10x button. Try it for yourself a few times:
If you enter a large number such as 35, the result will be too big to fit in the calculator's display, so it uses a compact notation that looks like 1e+35. That means the number is a one multiplied by 1035 (the e means exponent, and we'll see that exponents can be either positive or negative). Some numbers are even too big for that notation; see what happens when you calculate 10309. The calculator can't handle numbers that large, so it displays Infinity, which means a number so big that it's bigger than any number you can imagine.
Now let's examine what happens when we multiply two of these powers of 10 together:
| 102 × 103 | = | 100 × 1000 |
| = | 100,000 | |
| = | 105 | |
| = | 10(2+3) |
If you are having trouble following what we just did, I'll explain how you read these types of equations. The goal is to find out what the left side of the equation, 102 × 103, is equal to.
We know that 102 is 100 and 103 is 1000, so when you multiply the two powers together, the product is equal to 100 × 1000. That is shown in the first line of the equation.
But, you know what 100 × 1000 is; it is just 100,000, so the second line of the equation shows this. We don't repeat the left side of the equation since it does not change from line to line.
Furthermore, 100,000 is itself a power of 10, so we can write 100,000 as 105 which is shown by the third line of the equation above. Finally, the last line shows that the exponent 5 is just the sum of the two exponents in the first line of the equation.
Thus we have just shown that when we multiply two powers of 10 together, the answer is another power of 10 and the exponent is simply the sum of the two powers. In this example, 2 + 3 is 5, so 102 × 103 is 105.
Here's a crazy idea: can you have an exponent of 0? Let's multiply it by 100 and see what happens:
| 102 × 100 | = | 10(2+0) |
| = | 102 |
We ended up with 100 which is what we started with, so it appears that 100 is just 1. Verify that this is the case with a calculator:
It makes sense if you think about multiplying by a power of ten as adding zeros to the number. Multiplying by 1 doesn't add any zeros, so its exponent should be zero.
Now let's look at what happens when you divide two powers of 10:
| 105 ÷ 102 | = | 10 × 10 × 10 × 10 × 10 |
| 10 × 10 | ||
| = | 10 × 10 × 10 × 10 × 10 | |
| 10 × 10 | ||
| = | 10 × 10 × 10 | |
| = | 103 | |
| = | 10(5−2) |
The exponents subtract as you might have guessed. This is a really exciting result: multiplication and division can be performed using simple addition and subtraction!
What do you think will happen if we divide the top by a bigger number? Well we just found out that the exponents subtract when doing division, so we should end up with a negative exponent:
| 103 ÷ 105 | = | 10 × 10 × 10 | ||
| 10 × 10 × 10 × 10 × 10 | ||||
| = | 10 × 10 × 10 | |||
| 10 × 10 × 10 × 10 × 10 | ||||
| = |
|
|||
| = | 0.01 | |||
| = | 10−2 | |||
| = | 10(3−5) |
So what does it mean to have a negative exponent? One way to look at it is as we have been, where the exponent indicates how many zeros to add to a number when multiplying. If you multiply a number by 0.01, two zeros are taken away, so the exponent should be −2. A negative exponent can also be looked at as performing a reciprocal or inverse function:
| 10−2 | = | 1 |
| 102 |
Up to this point we have been working with a very limited collection of numbers, those that are integer powers of ten. But now let's think about what it would mean to have an exponent that is not a round number, such as 1/2 or 1/3. We've found that when we multiply by a number, the exponent tells us how many zeros to add, but what would it mean to add half a zero? Or a third?
That's a bizarre concept; how can you have a number with half of a zero? It seems impossible, but imagine what happens when you multiply two of them together:
101/2 × 101/2 = ???
Based on what we've seen happen with integer exponents when multiplying numbers together (the exponents add), we might guess that fractional exponents also add. If that were true, then the result above would be:
| 101/2 × 101/2 | = | 10(1/2+1/2) |
| = | 101 | |
| = | 10 |
Instead of thinking of exponents as adding zeros, with fractional exponents it makes more sense to think of them as adding powers of ten. Whatever 101/2 is, it gets us half way to another 10. You need two of them to get all the way there.
So what exactly is the value of 101/2? You might be tempted to say 5, since that is half of 10, but 5 is too much since 5 × 5 = 25, which is a lot more than 10. How about 4 then? 4 × 4 = 16, which is still too much. Three? 3 × 3 = 9, so we're getting closer, but 3 is not enough. We really need a calculator for this:
Now compare this to the square root of 10:
Another way to compute the square root of 10 should be:
If you've followed everything up to now, I have a surprise for you: you understand logarithms. Logarithms are just the exponents we used above. This table should look very familiar:
| Number | Logarithm | |
|---|---|---|
| 10 | 1 | |
| 100 | 2 | |
| 1,000 | 3 | |
| 10,000 | 4 | |
| 100,000 | 5 | |
| 1,000,000 | 6 |
The logarithm of a number tells you what exponent is needed to produce that number. If you raise 10 to that exponent, you get back the original number. Before there were calculators, people had to look up logarithms in a book. Here are the logarithms of a few numbers:
| Number | Logarithm | |
|---|---|---|
| 1 | 0.000 | |
| 2 | 0.301 | |
| 3 | 0.477 | |
| 4 | 0.602 | |
| 5 | 0.699 | |
| 6 | 0.778 | |
| 7 | 0.845 | |
| 8 | 0.903 | |
| 9 | 0.954 | |
| 10 | 1.000 |
Those numbers look scary, but the good news is that a calculator readily produces them for us with just a single button. But before we go on from this table, let's make a few observations.
If we add the logarithm of 2 to the logarithm of 5 we get:
| 0.301 |
| + 0.699 |
| 1.000 |
The result is the logarithm of 10. This is exactly what we should expect since logarithms of numbers add when we multiply the numbers together. Remember that a logarithm is just an exponent, and we know that exponents add when multiplying. In this case, 2 × 5 = 10, so the logarithm of 2 plus the logarithm of 5 should be the logarithm of 10, which is 1.
Let's see what we get if we add the logarithm of 4 and the logarithm of 5. We should expect to get the logarithm of 20:
| 0.602 |
| + 0.699 |
| 1.301 |
Is that what we got? If we take away the leading 1, we're left with the logarithm of 2 from the above table. And we know that multiplying by ten adds one to the logarithm, so we did in fact compute the logarithm of 20.
This is great. We can perform multiplication by adding logarithms together. Let's see
how to do this on a calculator by calculating 7 × 6 without using multiplication.
First compute the logarithm of 7; the calculator button we need is labeled log10.
Now add to that the logarithm of 6:
At this point we have the logarithm of the answer we are looking for, so we need to apply it as a power of 10:
You can see that there is some error in the calculation due to rounding by the calculator, but the result is very close to the expected answer of 42.
Now that you know the basics, to learn more please see the main Logarithms page. One of the things you will learn is how long it will take to double your money if it earns a certain amount of interest per year.