Textbook clarification: $\log = \ln$

Question

Textbook reads:

All logarithms are natural logarithms: $\log = \ln$.

Does this mean $n\log(n) = n\ln(n)$?

Answer 1

If the textbook is making clear that you should read $\log n$ to mean $\ln n$, then yes, barring any subscript for a base other than $e$, $n \log(n) = n\ln(n)$.

Answer 2

If $\log = \ln$, then yes, indeed, $n\log n = n\ln n$.

Note, however, that this is utterly unimportant if you are looking at it from the case of the big $O$ notation, which is highly likely, since

$$n\log n \in O(n\ln n)$$ and $$n\ln n \in O(n\log n)$$ or, in other words, $$O(n\log n)=O(n\ln n)$$

Answer 3

Yes. These notations are same. For example $\log x=\ln x$.

Answer 4

It just means that unless some other base is specified, the base of the logarithm is assumed to be $e$. That is, if you see $\log x$, the author means $\log_e x$, which can also be written $\ln x$.

The rule $\log_b x^n = n\log_b x$ holds for any base $b > 0$, with $b \neq 1$. In particular, it holds for $b = e$, so the answer to your question is yes.

Answer 5

Normally, the base of a logarithm must be specified as $\log_a$. A very common convention is $\ln \equiv \log_e$.

However, $\log$ without subscript can mean a few different things based on the context (and therefore must be always explicitly stated).

$\log \equiv \log_{10}$, is very common in many mathematical books and publications.

$\log \equiv \log_2$, in computer science.

$\log \equiv \ln$, in most physics and applied mathematical contexts.

Your textbook is in the latter category. It can still use other bases for logarithms, but unspecified means base $e$.