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Are $\log_{10}(x)$ and $\log_2(x)$ in the same big-O class of functions?
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Problem
Are $\log_{10}(x)$ and $\log_{2}(x)$ in the same big-O class of functions? In other words, can one say that $\log_{10}(x)=O(\log x)$ and $\log_{2}(x)=O(\log x)$?
Solution
Why?
$2^{\log_2 x} = x$
$\log_{10}(2^{\log_2 x}) = \log_{10} x$
$\log_2 x(\log_{10}2) = \log_{10} x$ $(*)$
$\log_2 x = \dfrac{\log_{10} x}{\log_{10}2}$
QED.
$() \log x^y = y\log x$
$2^{\log_2 x} = x$
$\log_{10}(2^{\log_2 x}) = \log_{10} x$
$\log_2 x(\log_{10}2) = \log_{10} x$ $(*)$
$\log_2 x = \dfrac{\log_{10} x}{\log_{10}2}$
QED.
$() \log x^y = y\log x$
Context
StackExchange Computer Science Q#4968, answer score: 2
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