Formulas and properties of logarithms

The logarithm of number b on the base a (log_a b) is defined as an exponent, in which it is necessary raise number a to gain number b (The logarithm exists only at positive numbers).

log_ab = x   if and only if   a^x = b

The following types of logarithms are exist

• log_a b - logarithm of number b on the base a (a > 0, a ≠ 1, b > 0)
• log b - common logarithms (Logarithms with base 10 are called common logarithms, a = 10).
• ln b - natural logarithms (Logarithms with base e are called natural logarithms, a = e).

Formulas and properties of logarithms

For any a; a > 0; a ≠ 1 and any x; y > 0.

1. a^log_ab = b
2. log_a 1 = 0
3. log_a a = 1
4. log_a(x · y) = log_ax + log_ay
5. log_a xy = log_ax - log_ay
6. log_a 1x = -log_ax
7. log_a x^p = p log_ax
8. log_a^k x = 1k log_a x,    for k ≠ 0
9. log_ax = log_a^c x^c
10. log_a x = log_b xlog_b a - change of base formula
11. log_a x = 1log_x a