\displaystyle{{\log}_{{{10}}}{\left({0.1}\right)}}=-{1} - or in other words, we take the 10 and flip flop it to the denominator of a fraction where we have \displaystyle\frac{{1}}{{10}} ...

remember that 0^0=1 so \log(0^0)=\log(1)=0. On the other hand \log(0) = undefined and thus so is 0 \log(0)

Press the keys, enter, then round it to four (see explanation below) You should get -0.0969 Explanation: 1) Find a scientific or graphics calculator & turn it on 2) press the \displaystyle{\log} ...

Of the many ways \log can be defined, one is \log x = \int_1^x\!\frac{1}{t}\,dt (I'm assuming you mean natural log, but if not, everything can be easily adapted, since \log_n x = \log x/\log n). ...

3×log10 can be written as log10^3=log1000 (Using identity: b×loga=loga^b) log1000 with base 3 will be the same as \frac {log1000}{log3}=log_{3}1000

I think that your difficulty comes from a confusion regarding what an "indeterminate form" is. Indeterminate forms show up in analysis via naive substitution when computing limits. For example, we ...