because

. You can say that 3 is the logarithm of 1,000 (for base 10), or, in mathematical

terms:

. Similarly, because

, you say that

. And because

, then

.

There can be logarithms to any base, but three bases occur frequently enough to have their own

nicknames:

Base-10 logarithms are called common logarithms.

Base-e logarithms are called natural logarithms.

Base-2 logarithms are called binary logarithms.

The logarithmic function naming is inconsistent among different authors, publishers, and

software writers. Sometimes Log means natural logarithm, and sometimes it means common

logarithm. Often Ln is used for natural logarithm, and Log is used for common logarithm. Names

like Log10 and Log2 may also be used to identify the base.

The most common kind of logarithm used in this book is the natural logarithm, so in this book

we always use Log to indicate natural (base-e) logarithms. When we want to refer to common

logarithms, we use

, and when referring to binary logarithms, we use

.

An antilogarithm (usually shortened to antilog) is the inverse of a logarithm. As an example of an

antilog, if y is the log of x, then x is the antilog of y. For another example, the base-10 logarithm of

1,000 is 3, so the base-10 antilog of 3 is 1,000.

Calculating an antilog is exactly the same as raising the base to the power of the logarithm.

That is, the base-10 antilog of 3 is the same as 10 raised to the power of 3 (which is

, or

1,000). Similarly, the natural antilog of any number is e (2.718) raised to the power of that

number. As an example, the natural antilog of 5 is

, or approximately 148.41.

Factorials and absolute values

So far we’ve covered mathematical operators that are written either between the two numbers, which

are the subject of the operation (such as the plus in 5 + 8), or before the number it operates on if there

is only one number (like the minus sign used as a unary operator described earlier, as in –5°). Next we

cover factorials and absolute values, which are mathematical operators that have a unique format in

typeset expressions.

Factorials

Although a statistical formula may contain an exclamation point, that doesn’t mean that you should

sound excited when you read the formula aloud (although it may be tempting to do so!). An

exclamation mark (!) after a number is shorthand for calculating that number’s factorial. To do that,