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:
$$\log a+\log b=\log ab$$
$$\log a-\log b=\log {\displaystyle \frac{a}{b}}$$
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 $$\log a-\log b=\log {\displaystyle \frac{a}{b}}$$.
So, $$\log 40-\log 4$$
$$\Rightarrow \log {\displaystyle \frac{40}{4}}$$
$$\Rightarrow \log 10$$
$$\log 10$$ 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
$${\log }_{10}10=1$$
$${\log }_{e}10=2.302$$