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I'm a bit confused, $\log_{10} x = \log x $ right? I believe I've read somewhere that $\log_{2} x = lg x$ but some people say lg = $\log$.

So what does lg really stand for? specifically when talking about "binary trees"

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There are many different notations. In the context of binary trees, you'd probably talk about $\log_2$, but I've seen it marked as $\log$, $\log_2$ or $\lg$. – Alfonso Fernandez Jan 19 '13 at 16:04
Almost certainly $\log_2$ $-$ the hint is in the word 'binary'! – Clive Newstead Jan 19 '13 at 16:05

$\lg$ will usually stand for $\log_2$ when talking about binary. In Germany and Russia, $\lg$ refers to $\log_{10}$. Source

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It is common that $\lg=\log_2$, but note that $\log_a = \Theta(\log_b)$, because $$\log_a x = \frac{\log_b x}{\log_b a}.$$

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