6.5 Common logarithms

When you see a root without a number in it, it is assumed to be a square root. That is, $\sqrt{\text{25}}$ is a shorthand way of writing $\sqrt[2]{\text{25}}$ . This rule is employed because square roots are more common than other types.

When you see a logarithm without a number in it, it is assumed to be a base 10 logarithm. That is, $\text{log}(\text{1000})$ is a shorthand way of writing ${\text{log}}_{\text{10}}(\text{1000})$ . A base 10 logarithm is also known as a “common” log.

Why are common logs particularly useful? Well, what is ${\text{log}}_{\text{10}}(\text{1000})$ ? By now you know that this asks the question “10 to what power is 1000?” The answer is 3. Similarly, you can confirm that:

We can also follow this pattern backward:

and so on. In other words, the common log tells you the order of magnitude of a number: how many zeros it has. Of course, ${\text{log}}_{\text{10}}(\text{500})$ is difficult to determine exactly without a calculator, but we can say immediately that it must be somewhere between 2 and 3, since 500 is between 100 and 1000.