After much thought, I have realized that x^0 = 1 is a joke on singles. It basically means you stay single (1) if you don’t multiply (x^0). :D

**Proof:**

- x^k = y [Read:
*You would have to multiply x successively k times to get y; k is called exponent*] - Inversely, k = log_x(y) [Read:
*You can divide y successively by x at most k times; the same k is called logarithm*] - 0 = log_x(1) [Read:
*At most how many times can you divide 1 successively by x? 0 because you can’t—1 is indivisible.*]

Therefore, raising x to the 0th power will give you 1 rewriting (3) according to (1) and (2) as: - x^0 = 1 [Read:
*How many times can I successively multiply x to get 1? 0. You stay single if you don’t multiply (multi = more than 1; multiply = become more than 1)—cheeky :D but that’s what it means.*]

**QED**.

**Another Proof:**

- x^m/x^n = x^(m-n)
*[exponentiation rules]* - a/a = 1
*[rules of arithmetic]* - When m == n,

1 = x^m/x^m = x^(m-m) = x^0*[by (1) and (2)]*

**QED.**

**Background:**

*A **logarithm** counts successive divisions* — it is the number of times you would have to divide a real number Y by another real number X until you cannot divide further (you reach 1; 1 is “indivisible”). Therefore, a logarithm basically answers the question: “how many times can I successively divide a number until it can no longer be divided?” ** Exponent** is another term for logarithm, but goes in the other

*direction*—

*it counts successive multiplications*. Logarithm is about

*reduction*while exponentiation is about

*growth.*Therefore, logarithms and exponentiation are mathematical

*inverses*:

log_x(x^y) = y

just as multiplication and division are inverses:

12 * 4 / 4 = 12

**Some Notation:**

log(x) [common log; base 10]

ln(x) [natural log; base *e*]

lg(x) [binary log; base 2]

floor(lg(x)) [floor binary log; base 2; greatest power of 2 less than the binary log of x]

ceil(lg(x)) [ceiling binary log; base 2; lowest power of 2 greater than the binary log of x]

etc.

**Examples:**

lg(8) = 3 [*I cannot divide 8 by 2 more than 3 times*].

2³ = 8