# Math equations and formulas

## Logarithms

The fundamental property of a logarithm (base b) is:

\$$\displaystyle y = b^x \iff x = log\_b y \iff b = \sqrt\[x   ] y \qquad \small{for} b \gt 1, y \gt 0

$$$
$$
log := log_{10} \\
lg := log_{2} \\
ln := log_{e}
$$$

Properties:

$$
log_{2}\ xy = log x + log y \\
log_{2}\ {x/y} = log x - log y \\
log_{2}\ x^y = y\ log\ x       \\
log_{2}\ 1 = 0              \\
log_{2}\ 2 = 1
$$

## Summations

$$
\displaystyle{
\sum_{i=1}^{n} \ i   = \frac {n(n+1)} {2}      \\
\sum_{i=1}^{n} \ i^2 = \frac {2n^3+3n^2+n} {6} \\
\sum_{i=0}^{n} \ 2^i = 2^{n+1} - 1             \\
\sum_{i=0}^{n} \ a^i = \frac {a^{n+1}-1} {a-1} \\
\sum_{i=1}^{n} \ i2^i = (n-1) 2^{n+1} + 2      \\
\sum_{i=1}^{n} \ \lceil log_2\ (i+1)\rceil = (n+1)\ log_2\ (n+1) - n  \\
\sum_{i=1}^{n} \ \lfloor log_2\ i \rfloor = (n+1)\ log_2\ (n+1) - 2n  \\
}
$$
