Question

How do you evaluate $$\log 0.001$$?

Answer 1

Hint: In the given question, we have been given an expression. This expression contains a function, which is the logarithm function. We have to solve the logarithm when we have been given the base and the argument of the logarithm. We do that by expressing the argument in terms of the given base and solving it by using the appropriate formula.

Formula Used:
We are going to use the formula of logarithm, which is:
$${\log }_{b}a=n\Rightarrow b^n=a$$

Complete step by step answer:
The given expression is $$p=\log 0.001$$. We have to solve for the value of $$p$$.
The basic formula of logarithm says,
If $${\log }_{b}a=n$$
then, $$b^n=a$$
In the question, we have not been given the value of $$b$$ (base). This is a standard representation. It is the same as the value of the base that we use in the normal day numbers – $$10$$.
Hence, putting $$b=10$$, $$a=0.001$$ and $$n=p$$, we get,
$${10}^p=0.001={\displaystyle \frac{1}{1000}}={\displaystyle \frac{1}{{10}^3}}$$
We know, $$a^{-b}={\displaystyle \frac{1}{a^b}}$$
Hence, $${10}^p={10}^{-3}$$
Thus, $$p=-3$$

Therefore, $$\log 0.001=-3$$

Additional Information:
The $$\log$$ function has other basic properties too:
$${\log }_x{x}^n=n$$
$${\log }_{a}b={\displaystyle \frac{1}{{\log }_{b}a}}$$

Note:
In the given question, we had been given a logarithmic expression. In this question, to solve for the answer, we needed to know the properties of the logarithmic function. We need to know the base of the number when nothing is given. We follow the standard base as used in the normal numbers – the base of Ten. We do that by expressing the argument in terms of the given base and solving it by using the appropriate formula. So, it is really important that we know the formulae and where, when, and how to use them so that we can get the correct result.