Question

How do you evaluate \[\log 0.003\]?

Answer 1

Hint: We will first simplify the decimal part and write it as a fraction. After that, we will use the Properties of Logarithm. In order to solve this we will use the Properties \[\log \left( {\dfrac{m}{n}} \right) = \log m - \log n\] and \[\log {m^n} = n\log m\]. After that we will solve the obtained expression by putting the log values. For that, we need to learn some log values or we can find them.

Complete step-by-step answer:
We need to evaluate \[\log 0.003\].
We know, \[0.003\] can be written as \[\dfrac{3}{{1000}}\].
Writing \[0.003\] as \[\dfrac{3}{{1000}}\] in \[\log 0.003\], we get
\[\log 0.003 = \log \left( {\dfrac{3}{{1000}}} \right)\]
Now, using the Property \[\log \left( {\dfrac{m}{n}} \right) = \log m - \log n\], we get
\[\log 0.003 = \log \left( {\dfrac{3}{{1000}}} \right) = \log 3 - \log 1000\]
Now, \[1000\] can be written as \[{10^3}\].
Writing \[1000\] as \[{10^3}\], we get
\[\log 0.003 = \log \left( {\dfrac{3}{{1000}}} \right) = \log 3 - \log 1000\]
\[ = \log 3 - \log {10^3}\]
Now, using the property \[\log {m^n} = n\log m\], we get
\[\log 0.003 = \log \left( {\dfrac{3}{{1000}}} \right) = \log 3 - \log 1000\]
\[ = \log 3 - \log {10^3}\]
\[ = \log 3 - 3\log 10\]
Now, we know \[\log 10 = 1\]
Using \[\log 10 = 1\] in the above expression, we have
\[\log 0.003 = \log \left( {\dfrac{3}{{1000}}} \right) = \log 3 - \log 1000\]
\[ = \log 3 - \log {10^3}\]
\[ = \log 3 - 3\log 10\]
\[ = \log 3 - 3\left( 1 \right)\]
\[ = \log 3 - 3\]
\[ \Rightarrow \log 0.003 = \log 3 - 3 - - - - - - (1)\]
Using this we can evaluate \[\log 3\] or we know the value of \[\log 3\].
We know, \[\log 3 = 0.4771\] (approx)
Now, substituting the value of \[\log 3\] in (1), we get
\[ \Rightarrow \log 0.003 = 0.4771 - 3\]
\[ \Rightarrow \log 0.003 = - 2.5229\]
Hence, we get
\[\log 0.003 = - 2.5229\]
So, the correct answer is “ - 2.5229”.

Note: We could have calculated the value of \[\log 3\] but we should be aware of some log values. We could have made a mistake while using the Logarithmic property \[\log {m^n} = n\log m\]. We usually apply this property when we are given \[{\left( {\log m} \right)^n}\] but we need to see that \[{\left( {\log m} \right)^n}\] and \[\log {m^n}\] are different and we cannot apply the property when we are given \[{\left( {\log m} \right)^n}\].