$lo g_{b} x = y ⟺ b^{a} = y$

Base: $0 < b \neq = 1$
$0 < x$

Logarithmic identities

Identity	Formula
Log of the Base	$lo g_{b} b = 1$
Log of Product	$lo g_{b} (x y) = lo g_{b} x + lo g_{b} y$
Log of Quotient	$lo g_{b} (x / y) = lo g_{b} x - lo g_{b} y$
Log of Power	$lo g_{b} x^{n} = n lo g_{b} x$
Log of One	$lo g_{b} 1 = 0$
Log of the nth Root	$lo g_{b} (n x) = \frac{lo g _{b} x}{n}$
Change of Base	$lo g_{b} x = \frac{lo g _{c} x}{lo g _{c} b}$
Reciprocal of Logarithm	$lo g_{b} x = \frac{1}{lo g _{x} b}$
Power of a Log	$x^{l o g_{b} y} = y^{l o g_{b} x}$
Logarithm of Reciprocal	$lo g_{b} (1/ a) = - lo g_{b} a$
	$lo g_{b} b^{x} = x$	( $x \in R$ )
	$b^{l o g_{b} x} = x$
	$b^{r l o g_{b} x} = x^{r}$

also see limits of logarithmic functions

Binary logarithm

$lo g_{2} n = l g n$
$(l g n)^{l g n} = n^{l g l g n}$
$2^{l g l g n} = l g n$
$2^{l g n} = n^{l g 2} = n^{1} = n$
$4^{l g n} = n^{l g 4} = n^{2}$
$4^{l g n} = n^{2}$
$n^{1/ l g n} = 2$
$n^{1/ l g n} = 2$
$(n /2) l g (n /2) \leq l g (n!) \leq n l g n$
- $⟹ l g (n!) = Θ (n l g n)$ ^[see CRLS: t8.1, e8.1-2]

Notation

$l g^{k} n = (l g n)^{k}$

$l g^{(k)} n$

$l g l g n = l g (l g n) = l g^{(2)} n$

Natural Logarithm

$ln x := lo g_{e} (x)$
$ln (1) = 0$
$ln (e) = 1$
$ln (e^{x}) = x$
$e^{l n x} = x$
$e^{r l n x} = x^{r}$