Powers, Roots, and Logarithms for Use in Biostatistics

These three mathematical operations — working with powers, roots, and logarithms — are all related to the idea of repeated multiplication. These basic functions are used to help build more complex formulas.

Raising to a power

Raising to a power is a shorthand way to indicate repeated multiplication. You indicate raising to a power by

• Superscripting in typographic formulas, such as 5³ = 125

• ** in plain text formulas, such as 5**3 = 125

• ^ in plain text formulas, such as 5^3 = 125

All the preceding expressions are read as "five to the third power," or "five cubed," and tell you to multiply three fives together: 5 × 5 × 5, which gives you 125.

These statements about powers are true, too:

• A power doesn't have to be a whole number. You can raise a number to a fractional power. You can't visualize this in terms of repeated multiplications, but your scientific calculator can show you that 2.6^3.8 is equal to approximately 37.748.

• A power can be negative. A negative power indicates the reciprocal of the quantity: x^–1 really means 1/x, and, in general, x^–n is the same as 1/xⁿ.

Almost every time you see e used in a formula, it's being raised to some power. It's almost as if e were born to be raised to powers. It's so common that raising e to a power (that is, to some exponent) is called exponentiating, and another way of representing e^x in plain text is exp(x).

And x doesn't have to be a whole number: Using any scientific calculator or spreadsheet, you can show that exp(1.6) equals 4.953 (approximately).

Taking a root

Taking a root involves asking the power question backwards: "What base number, when raised to a certain power, gives some specific number?" For example, "What number, when squared, gives 100?" Well, 10 × 10, or 10², gives 100, so the square root of 100 is 10. Similarly, the cube root of 1,000,000 is 100 because 100 × 100 × 100, or 100³, is a million.

Root-taking is indicated by a radical sign in a typeset formula, where the entire thing to be square-rooted is located "under the roof" of the radical sign, as shown here:

You indicate other roots by putting a number in the notch of the radical sign. For example, because 2⁸ is 256, the eighth root of 256, or

You also can indicate root-taking by using the fact (from algebra) that

or as x^(1/n) in plain text.

Using logarithms

In addition to root-taking, another way of asking the power question backwards is "What exponent (or power) must you raise a certain base number to in order to get some specified number?" The distinction between roots and logarithms is this: for root-taking, you specify the power and ask for the base; for logarithms, you specify the base and ask for the power (or exponent).

For example, "What power must you raise 10 to in order to get 1,000?" The answer is 3 because 10³ = 1,000. You can say that 3 is the logarithm of 1,000 (for the base 10), or, in mathematical terms: Log₁₀(1,000) = 3. Similarly, because 2⁸ = 256, you say that Log₂(256) = 8. And because e^1.6 = 4.953, then Log_e(4.953) = 1.6.

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. They were commonly used (no pun intended) in the old days (before calculators) because they actually made numerical calculations easier. To multiply two large numbers together, you could add their logarithms, then find the antilog of the sum.

• 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.

An antilogarithm (usually shortened to antilog) is the inverse of a logarithm — if y is the log of x, then x is the antilog of y. For 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 10³, or 1,000). Similarly, the natural antilog of any number is just e (2.718) raised to the power of that number: The natural antilog of 5 is e⁵, or 148.41, approximately.