
ln(x) log(x) =  ln(10) 
ln(x) lg(x) =  ln(2) 
The base2 log function is frequently used in the analysis of computer algorithms
ln(a.b) = ln(a) + ln(b) ln(a/b) = ln(a)  ln(b) ln(1/b) = ln(b) ln(a^{n}) = n ln(a) 
1 1 1 1 H_{N} = 1 +  +  +  + ... +  2 3 4 N 
Examples:
H_{1} = 1 1 H_{N} = 1 +  = 1.5 2 1 1 H_{N} = 1 +  +  = 1.8333333... 2 3 
The value of H_{N} is equal to the area of the squares from 1 to N
The value of ln(N) is equal to the area under the graph of f(x)=1/x from 1 to N+1
1 H_{N} ~= ln(N) + γ +  12N γ ~= 0.57721... (Euler's constant) 
F_{0} = 1 F_{1} = 1 F_{N} = F_{N1} + F_{N2} for N >= 2 
Examples:
F_{1} = 1 F_{2} = 1 F_{3} = F_{2} + F_{1} = 1 + 1 = 2 F_{4} = F_{3} + F_{2} = 2 + 1 = 3 F_{5} = F_{4} + F_{3} = 3 + 2 = 5 
 F_{N+1} 1 + \/ 5  > φ =  F_{N} 2 
φ^{N} F_{N} ~=  (rounded to the nearest integer)  \/ 5 φ ~= 1.61803... (the "Golden ratio"  See: click here) 
0! = 1 N! = 1 × 2 × 3 × .... × N for N >= 1 
Examples:
0! = 1 1! = 1 2! = 1 × 2 = 2 3! = 1 × 2 × 3 = 6 4! = 1 × 2 × 3 × 4 = 24 
N + +  N  / N! ~=    / 2 π N  e  \/ + + where: π ~= 3.1415926535... e ~= 2.718281828... (Base of the natural log function) 
Proof:

Proof:

Proof:

One such serie sum is:
1 + 2 + 3 + .... + (n1) + n 
1 + 2 + 3 + .... + (n2) + (n1) + n Regroup the items as follows: (1+n) + (2+(n1)) + (3+(n3)) + .... 
1 + x + x^{2} + x^{3} .... + x^{n} 
S = 1 + x + x^{2} + x^{3} .... + x^{n} Multiply both side by x: x S = x (1 + x + x^{2} + .... + x^{n}) = x + x^{2} + x^{3} .... + x^{n+1} 