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1. (mathematics) For a number ${\displaystyle x}$, the exponent by which a given base number must be raised in order to obtain the power ${\displaystyle x}$. Written ${\displaystyle \log _{b}x}$. For example, ${\displaystyle \log _{10}1000=3}$ because ${\displaystyle 10^{3}=1000}$ and ${\displaystyle \log _{2}16=4}$ because ${\displaystyle 2^{4}=16}$.