# Bayes Information Criterion — what does log mean?

Super basic question about the BIC — is it defined in terms of log base ten or the natural logarithm? I see the latter on Wikipedia; but see ‘log’ not ‘ln’ in the original paper (though am aware that ‘log’ can mean ‘ln’...)

• Here $\log$ means the natural log, logarithmus naturali, or ln. Feb 19 at 16:22
• Also related: stats.stackexchange.com/questions/205312/… Feb 19 at 16:37

Note that $$\log_b(x)=\frac{\log_a(x)}{\log_a(b)} =: c \log_a(x)$$. In that sense, all logs are proportional. So they all have the same order. In statistics, the logarithms are often not distinguished for this reason. I saw people use “log” for cases where they don’t care about the base (as it just leads to a different constant).
In this specific case, using a different base will change the result. You’re minimizing something $$+ c \ln(d)$$, where $$c$$ is a constant which is determined by the basis of choice. So, if you choose a different basis, the penalty becomes larger (or smaller).