# Basic properties of logarithms

The logarithm of a given number b is the exponent to which another fixed number, the base a, must be raised, to produce that number b $(a>0, \ a\neq 1, \ b>0)$.

$$\displaystyle \log_ab=c, \ a^c=b$$## Logarithmic identities

Logarithm power rules are very useful whenever expressions involving logarithmic functions need to be simplified: $$\displaystyle \alpha \log _ax=\log _a x^{\alpha}$$ $$\frac{1}{\alpha}\log _ax=\log _{a^{\alpha}}x$$ $$\displaystyle a^{\log_ax}=x$$ The base of logarithm can be changed in other. It's very helpful, when we need to calculate logarithm with calculator which has only standard 10 and e bases. Change of base formula: $$\displaystyle \log_ax=\frac{\log_bx}{\log_ba}$$ Sum and difference of two logarithms with the same base properties (or product to sum and quotient to difference) formulas: $$\log_ax+\log_ay=\log_axy$$ $$\log_ax-\log_ay=\log_a\frac{x}{y}$$

2020-11-28

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