Review of Exponentiation

Exponentiation is the process of raising a base to a power, commonly denoted x^y. There are many ways of saying this, the most common form is: "x to the y" which comes from the longer way of saying it when y is an integer n: "x to the n-th power."

In the beginning, exponentiation was defined as iterated multiplication. Iteration is the process of repeating a function several times, so this would be a function x*x*x*...*x where there are n x's. This forms the basis of exponentiation for integers, but what about real numbers? To define exponentiation for real numbers, mathematicians had to find certain rules that were satisfied by integer exponentiation, and use them to generalize it to a continuously iterated multiplication. The rules that were found centuries ago are now known as the laws of exponentiation:

(`x`^m)(`x`ⁿ) = `x`^{(m + n)}	additive property of powers
(`x`^m)ⁿ = `x`^(mn)	multiplicative property of powers

These laws allow exponentiation to be extended to rational numbers fairly easily, and from there to real numbers by choosing rational numbers that are closer and closer to a real number you want to use. For example, if you wanted to find x^(3/5), then you could use the multiplicative property, and notice that: (x^(3/5))⁵ = x³. So x^(3/5) is the value y that satisfies: y⁵ = x³, because (x^(3/5))⁵ = x^(3/5)*5 = x³.

Extending the definition of exponentiation to real numbers allows inverse functions of exponentiation, namely roots and logarithms. Eventually, after the investigation of logarithms, the same rules are found to apply to them as well. These rules have become known as the laws of logarithms:

log_x(`mn`) = log_x(`m`) + log_x(`n`)	additive property of logarithms
log_x(`m`ⁿ) = `n` log_x(`m`)	multiplicative property of logarithms

These properties form the basis of exponential and logarithmic arithmetic and algebraic manipulations. There are sometimes two additional laws that are based on each of the two above, but they are more of a short-cut, because they can be derived from the properties above:

(`x`^m)/(`x`ⁿ) = `x`^{(m - n)}	subtractive property of powers
log_x(`m`/`n`) = log_x(`m`) - log_x(`n`)	subtractive property of logarithms

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