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# Formulas and properties of logarithms

Definition
The logarithm of number b on the base a (loga 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).

logab = x   if and only if   ax = b

## The following types of logarithms are exist

• loga 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. alogab = b
2. loga 1 = 0
3. loga a = 1
4. loga(x · y) = logax + logay
5. loga xy = logax - logay
6. loga 1x = -logax
7. loga xp = p logax
8. logak x = 1k loga x,    for k ≠ 0
9. logax = logac xc
10. loga x = logb xlogb a - change of base formula
11. loga x = 1logx a