# 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
Logarithms of a powers: $\displaystyle \alpha \log _ax=\log _a x^{\alpha}$,  $\frac{1}{\alpha}\log _ax=\log _{a^{\alpha}}x$,  $\displaystyle a^{\log_ax}=x$
Change of base property: $\displaystyle \log_ax=\frac{\log_bx}{\log_ba}$
Logarithms of a product and quotient: $\log_ax+\log_ay=\log_axy$,  $\log_ax-\log_ay=\log_a\frac{x}{y}$

Last time edited 2019-01-19

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