# What does $\lg x$ mean? is it $\log_2 x$ or $\log_{10} x$ in binary trees

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"

• 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 $\lg$. Commented Jan 19, 2013 at 16:04
• Almost certainly $\log_2$ $-$ the hint is in the word 'binary'! Commented Jan 19, 2013 at 16:05

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

• In Hungary as well Commented Mar 7, 2018 at 17:26

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

• This remark is interesting because for the most part algorithms evaluation do not care about multiplicative factors.
– zwim
Commented Sep 27, 2021 at 11:14

The ISO 80000 specification, published in 2019, tries to resolve the numerous ambiguities in the notations for logarithms. It recommends the following notations:

$$\lg x\equiv\log_{10}x\\ \ln x\equiv\log_e x\\ {\rm lb}\, x\equiv\log_2 x$$.

These ISO recommendations try to avoid the ambiguity in the notations for the common logarithm $${\rm Log}\, x\equiv\log x\equiv\log_{10} x$$, as the same notation is sometimes used for the natural logarithm $$\log x\equiv\ln x\equiv\log_e x$$. The ISO rules also discourage the use of $${\rm lg}\, x\equiv\log_2 x$$. See a discussion of the ambiguities HERE.