I am having trouble distinguishing between the symbol In and the word log.
(1) What's the difference between the two?
(2) What exactly does the symbol In mean?
It's actually $\displaystyle \ln,$ which means "natural logarithm."
The natural logarithm can be rewritten as $\displaystyle \log_e,$ where $\displaystyle e$ is a constant. In particular, most calculus books treat $\displaystyle \log$ equivalently as $\displaystyle \ln.$
($\displaystyle \log$ has the base 10, but there's no need to write it.)
Log, unless otherwise specified, is usually considered to be base 10. Ln is simply a notation. Ln is actually log base e.
$\displaystyle \log{10} = \log_{10}{10} = 1$
$\displaystyle \log_e{e} = \ln{e} = 1$
Ln is an abbreviation for logarithmus naturalis. Read more here if you're interested:
Natural logarithm - Wikipedia, the free encyclopedia
Firstly, if you're talking about the logarithm, its ln (L and N) instead of In
1. ln is always the natural logarithm, (logarithm to the base e). But log can mean either the natural logarithm or logarithm to the 10.
2. ln means either logarithmus naturalis or logarithm napieren, I'm not sure about this.