I watched a lecture, and it included the following segment.
There are two functions: $f(n)$ and $g(n)$.
Let's say we want to find any asymptotic relationship between those functions.
We do a logarithm to each function, and we find out that:
$\log(f(n)) < o(\log(g(n))$ - a small o.
When we have $f(n)$ and $g(n)$ in their normal form, an "o" asymptotic relationship will mean:
$f(n) < c \cdot g(n)$
but for the logarithm function it is possible to write:
$\log (f(n)) < \log(g(n) + c$
and I am confused about the $+c$ $\hspace{0.0cm}$ part. Where did it come from?
I tried to think of $\log(c \cdot g(n)) = \log(c) + \log(g(n))$
but we can't say $\log(c) = c$, $\hspace{0.0cm}$ aren't we?
Thanks for reading.
$C$ is an "arbitrary" constant. This means that it has a fixed value, but we don't know what it is. Oftentimes people will refer to constant transformations of $C$ as still $C$, as long as it creates no ambiguity. So, $\log(C)$ is still an arbitrary constant, and, if we have $\log(C)$ we no longer have any use for $C$. So, it does become a different arbitrary constant. However, since we have no use for the "old" $C$, we just re-use its name.
More properly, it should be written as:
$$\log(f(n)) < \log(C_1\cdot g(n))\\ \log(f(n)) < \log(g(n)) + \log(C_1) \\ C_2 = \log(C_1) \\ \log(f(n)) < \log(g(n)) + C_2$$
However, since $C_1$ is no longer used after this, many mathematicians just keep on trucking with the original $C$.