Exponents
An exponent is a "power number." In many respects, it is related to multiplication. If you raise a number to the second power, the exponent would be "2." For instance, three raised to the second power is 3 x 3, which equals "9." This would be written,

3² = 9

For which you would say, "three to the two equals nine." The numeral "3" is many times called the "base," while "2" is called the "exponent," or "power." Because the exponent "2" is many times used in describing physical area, it is also called the "square" of a number. Our example could be restated as "three squared."

Related to the powers of numbers are their roots. Taking the square root of a number is the same as,

²√3 = 3^(1/2) = 3^0.5 = 1.73205... (approximately)

and, conversely, — 1.73205... x 1.73205... = 3

As you can see, with squares and square roots, the exponents are opposites, or reciprocals — "2" and "1/2". With roots, the exponent is a fraction.

"Base" can also refer to a system of numbers. We are most familiar with "base 10" numbers — a system with numerals 0 - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9. As we all know, when incrementing through this series of numerals, once you get to the end of the allotted set of symbols (those representing the numerals of our base 10 system), you start back at "1" in the "tens" place, and use a "0" to indicate an empty "ones" place. And, at "99," adding one more yields a "1" in the "hundreds" place, a "0" in the "tens" place, and another "0" in the "ones" place. Of course, the same procedure would apply to other number systems.

Systems of numbers can be founded on any base. Most familiar to the field of computers are the bases "2" and "16." For base "2," or "binary," the only significant numeral is "1." Zero ("0") is only a place holder, meaning that the place within the numeral is empty. To count in binary, the first few numbers would be, 1, 10, 11, 100, 101, 110, 111,... This is equivalent to the decimal (base 10) count of "1" through "7."

In base "16," also called "hexadecimal" or simply "hex," we need more than the symbols 1 - 9. Letters of the alphabet were chosen to fill in the additional symbols needed to describe base "16" — 0 - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - A - B - C - D - E - F.

In binary, or base 2, the following numbers, 1, 10, 100, 1000, would be in decimal, 1, 2, 4, 8. These could also be written in terms of powers of base 2 as, 2⁰, 2¹, 2², 2³.

Base 10 exponents come in handy to make very large or very small numbers easier to read and understand. For instance, the distance to our closest neighboring star system is 41,039,810,000,000 kilometers. This number would be easier to read as 4.104 x 10¹³, sometimes written 4.104e13 (where "e" stands for exponent of base 10).

Logarithms
Logarithms, or "logs," are simply the exponents used to describe a resulting number. For instance,

10² = 100

The numeral "2" is the exponent of "10" used to get the result "100." Restated, "2" is the logarithm of "100," when using a base of "10." Fractional exponents can result in other numbers like,

10^0.4771212547 = 3 (approximately)

Here, "0.4771212547" is the approximate logarithm of "3" when using base "10."

A useful conversion can be performed when you have a logarithm in one base, and desire the logarithm in a different base. Perhaps the most widely used base is called "e," or the base of the natural system of logarithms, approximately equal to "2.718...". Say you have a calculator that gives you only the natural logs of numbers. To calculate the desired log, say base "10," simply divide the current log of a number by the log of "10." Natural logarithm functions are sometimes written with ln, and sometimes Log_e. An example of such a conversion would be,

Log_e(3) / Log_e(10) = Log₁₀(3) = 0.4771212547196624372950279032551... (approximately).

An example of logarithm use can be found in the discussion of iron abundance in stars.