Question

How do you condense \[\log 40-\log 4\]?

Answer 1

Hint: Logarithms have special properties associated and calculations are carried in a unique way. There are some formulae for their calculation which are as follows:
\[\log a+\log b=\log ab\]
\[\log a-\log b=\log \dfrac{a}{b}\]
Using these formulae, we can solve most logarithm questions without actually using logarithm tables.

Complete step by step answer:
According to the question we have to condense \[\log 40-\log 4\] which means we have to simplify as much as possible
We will solve it part by part, firstly we will take \[\log 40\]
Using the above formula we will simplify \[\log 40\],
\[\log 40\] can be written as \[\log (4\times 10)\] or \[\log (5\times 8)\], the selection will depend on whether the factors will help simplify the expression or not. If we take \[\log (4\times 10)\], then using the logarithm formula we can expand it. It has factors which can help in simplifying the expression whereas \[\log (5\times 8)\]will add more to the complexity of the expression. So we will use \[\log (4\times 10)\].
So, \[\log 40=\log (4\times 10)\]
Using the formula, \[\log ab=\log a+\log b\]
We have, \[\log (4\times 10)=\log 4+\log 10\]
Now, taking the other part of the expression that is \[-\log 4\].
We can now see clearly that the simplification of \[\log 40\] which has \[\log 4\]as a result expansion using the logarithm formula can be cancelled using the \[-\log 4\] from the remaining equation.
So the choice of factor we chose is favourable for us in simplifying the given expression.
We have,
\[\log 40-\log 4\]
\[\Rightarrow (\log 4+\log 10)-\log 4\]
\[\log 4\] will get cancelled and what is left is the simplified version of the expression
\[\Rightarrow \log 10\]
Now, depending upon the base of the logarithm whether it 10 or ‘e’, the value will differ, that is
\[{{\log }_{10}}10=1\]
\[{{\log }_{e}}10=2.302\]

Note: Logarithm function should be dealt carefully. Also the expression can be simplified using another logarithm formula which is \[\log a-\log b=\log \dfrac{a}{b}\].
So, \[\log 40-\log 4\]
\[\Rightarrow \log \dfrac{40}{4}\]
\[\Rightarrow \log 10\]
\[\log 10\] is the simplified or the condensed form. And depending upon the base of the logarithm whether it 10 or ‘e’, the value will differ, that is
\[{{\log }_{10}}10=1\]
\[{{\log }_{e}}10=2.302\]