Formulas and properties of logarithms.
Definition 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_{a}b = 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.
- a^{logab} = b
- log_{a} 1 = 0
- log_{a} a = 1
- log_{a}(x · y) = log_{a}x + log_{a}y
- log_{a} xy = log_{a}x - log_{a}y
- log_{a} 1x = -log_{a}x
- log_{a} x^{p} = p log_{a}x
- log_{ak} x = 1k log_{a} x, for k ≠ 0
- log_{a}x = log_{ac} x^{c}
- log_{a} x = log_{b} xlog_{b} a - change of base formula
- log_{a} x = 1log_{x} a
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