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As for “Log in to host.com” versus “Log into host.com,” I would use the former because I think that “log in” is a fixed phrase. Martha’s answer to another question is also related. Added : The Corpus of Contemporary American English (COCA) lists 65 occurrences of “log in to” and 58 occurrences of “log into,” both including ...
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log(x) = ln(x)/ln(10) via the change-of-base rule, thus the Taylor series for log(x) is just the Taylor series for ln(x) divided by ln(10). $\endgroup$ – correcthorsebatterystaple Commented Mar 18 at 14:35
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$2^{\log_2(3)} = 3$. Do any of those appear to be equal? (Whenever you are wondering whether some general algebraic relationship holds, it's a good idea to first try some simple numerical examples to see if it is even possible.) Actually, the only way that $(\log_2(3))^2 = 2 \log_2(3)$ could hold is if $\log_2(3)$ were equal to 2 or 0.
For my money, log on to a system or log in to a system are interchangeable, and depend on the metaphor you are using (see comment on your post). I suppose there is a small bit of connotation that "log on" implies use, and "log in" implies access or a specific user.
$\begingroup$ 0/0, log1(1) are the same, and tan(90), sin(90)/cos(90) and a/0 are the same. They claim it is undefined due to division but I beg to differ.
$\begingroup$ As an aside, to make matters worse, some authors will write $\log$ without a subscript and mean different things than one another.
There is currently no well-known function $\;f(x,y)\;$ such that $\;\log(x)\cdot\log(y)=\log(f(x,y)).\;$ That is, the function $\;f(x,y):=x^{\log(y)}=y^{\log(x)}\;$ has not been given a name yet, although it is a valid function. This situation may change at some future time. There are comparatively few named functions but new ones appear sometimes.
Note that $$\frac{1}{1+x}=\sum_{n \ge 0} (-1)^nx^n$$ Integrating both sides gives you \begin{align} \ln(1+x) &=\sum_{n \ge 0}\frac{(-1)^nx^{n+1}}{n+1}\\ &=x-\frac{x^2 ...