Exponents (Math)

Definitions

In a^b=c, a is a multiplied by itself b times.
a^b is almost never b^a.
Exponents have two inverse functions logarithms (log, lg, ln) (logarithms are the purest inverse of exponents) and roots (√) (a.k.a. radicals):

* ^b√c = a
* log_a c = b

if b = 0:: $a^b\; =\; 1\; ,$, but undefined if a = 0

if b > 0 and b is a whole:: $a^b\; =\; ,$ a multiplied by itself b times $=\; prod\_\{n\; =\; 1\}^\{b\}\; a\; ,$: $a^\{-b\}\; =\; frac\{1\}\{a^b\}\; ,$, so a^-b is here undefined if a = 0

if b is not both whole and real:: $a^b\; =\; e^\{b\; ln\; a\}\; ,$

Roots can be defined by:
$a^\{1/b\}\; =\; sqrt\; [b]\; \{a\}\; ,$

e^x (the x:th power of e, e = "Euler's Number" and is the natural number) can be defined by:
$e^x\; =\; 1\; +\; sum\_\{n\; =\; 1\}^\{infty\}\; frac\{x^n\}\{n!\}\; =\; 1\; +\; sum\_\{n\; =\; 1\}^\{infty\}\; frac\{x^n\}\{\; prod\_\{k\; =\; 1\}^\{n\}\; k\; \}\; ,$

e^x has an inverse function, ln x (the natural logarithm), which can be defined by: (infinity serie)
Let "f"(k) be $k\; +\; frac\{(x\; -\; 1)\; *\; lceil\; k\; /\; 2\; ceil\; ^2\}\{f(k\; +\; 1)\}\; ,$, where x should be known
$ln(x)\; =\; f(0)\; ,$

$log\_a\; x\; ,$ is defined for all a ≠ 0 and a ≠ 1.

Ln is always the natural logarithm (e-base).
Lg and Log is always used as the common logarithm (10-base) if no base is written.
Lg is most commonly used for 10-base otherwise Log is more common.

What we can define with exponents

$i\; =\; sqrt\{-1\}\; ,$

$pi\; =\; frac\{ln(-1)\}\{i\}\; ,$

$|a|\; =\; sqrt\{a^2\}\; ,$ (gives a possitive for real numbers and is called the absolute value for a)

$sng(a)\; =\; frac\{a\}\; ,$, but if a = 0, sgn(a) = 0 (sgn is called signum)

("a" shifted to left "b" times)
$a\; >>\; b\; =\; a\; /\; 2^b\; ,$ ("a" shifted to right "b" times)

Laws

$a^x\; *\; a^y\; =\; a^\{x\; +\; y\}\; ,$

$frac\{a^x\}\{a^y\}\; =\; a^\{x\; -\; y\}\; ,$

$(a^x)^y\; =\; a^\{xy\}\; ,$ in general, but calculate "a"^"x" first then raise it to "y"
-a^b = -(a^b) ≠ (-a)^b

$e^\{ln\; a\}\; =\; a\; ,$
$ln\; e^a\; =\; a\; ,$
$ln\; (ab)\; =\; ln\; a\; +\; ln\; b\; ,$

$ln\; (frac\{a\}\{b\})\; =\; ln\; a\; -\; ln\; b\; ,$

$ln\; (a^b)\; =\; b\; ln\; a\; ,$
Polynoms are commonly used as functions (1 input value for a function most have exact 1 output value). A polynoms grade is chosen by the highest exponent on x, all exponent must be natural numbers. A x-grade polynom's structure is
$k\_0\; sum\_\{n=1\}^\{x\}\; k\_n\; x^n\; ,$A 2:nd grade polynom is called a binom, and a 3:th grade polynom is called a trinom.

(a + b)² = a² + 2ab + b²
Proof:
: $(a\; +\; b)^2\; =\; (a\; +\; b)(a\; +\; b)\; =\; aa\; +\; ab\; +\; ba\; +\; bb\; =\; a^2\; +\; ab\; +\; ab\; +\; b^2\; =\; a^2\; +\; 2ab\; +\; b^2\; ,$

(a + b)(a - b) = a² - b²
Proof:
: $(a\; +\; b)(a\; -\; b)\; =\; aa\; +\; ab\; -\; ba\; -\; bb\; =\; a^2\; +\; ab\; -\; ab\; -\; b^2\; =\; a^2\; -\; b^2\; ,$

$x^2\; +\; kx\; =\; m\; Rightarrow\; x\; =\; -\; frac\{k\}\{2\}\; pm\; sqrt\{(frac\{k\}\{2\})^2\; -\; m\}\; ,$.

See also

Trigonometry

Sources

http://mathworld.wolfram.com/
Matematiklexikon "(Swedish dictionary for math)"
Matematik 3000 "(Swedish schoolbook for math)"