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Question: How do you evaluate \[\log 0.003\]?...

How do you evaluate log⁡0.003\log 0.003?

Explanation

Solution

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⁡(mn)=log⁡m−log⁡n\log \left( {\dfrac{m}{n}} \right) = \log m - \log n and log⁡mn=nlog⁡m\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\log 0.003.
We know, 0.0030.003 can be written as 31000\dfrac{3}{{1000}}.
Writing 0.0030.003 as 31000\dfrac{3}{{1000}} in log⁡0.003\log 0.003, we get
log⁡0.003=log⁡(31000)\log 0.003 = \log \left( {\dfrac{3}{{1000}}} \right)
Now, using the Property log⁡(mn)=log⁡m−log⁡n\log \left( {\dfrac{m}{n}} \right) = \log m - \log n, we get
log⁡0.003=log⁡(31000)=log⁡3−log⁡1000\log 0.003 = \log \left( {\dfrac{3}{{1000}}} \right) = \log 3 - \log 1000
Now, 10001000 can be written as 103{10^3}.
Writing 10001000 as 103{10^3}, we get
log⁡0.003=log⁡(31000)=log⁡3−log⁡1000\log 0.003 = \log \left( {\dfrac{3}{{1000}}} \right) = \log 3 - \log 1000
=log⁡3−log⁡103= \log 3 - \log {10^3}
Now, using the property log⁡mn=nlog⁡m\log {m^n} = n\log m, we get
log⁡0.003=log⁡(31000)=log⁡3−log⁡1000\log 0.003 = \log \left( {\dfrac{3}{{1000}}} \right) = \log 3 - \log 1000
=log⁡3−log⁡103= \log 3 - \log {10^3}
=log⁡3−3log⁡10= \log 3 - 3\log 10
Now, we know log⁡10=1\log 10 = 1
Using log⁡10=1\log 10 = 1 in the above expression, we have
log⁡0.003=log⁡(31000)=log⁡3−log⁡1000\log 0.003 = \log \left( {\dfrac{3}{{1000}}} \right) = \log 3 - \log 1000
=log⁡3−log⁡103= \log 3 - \log {10^3}
=log⁡3−3log⁡10= \log 3 - 3\log 10
=log⁡3−3(1)= \log 3 - 3\left( 1 \right)
=log⁡3−3= \log 3 - 3
⇒log⁡0.003=log⁡3−3−−−−−−(1)\Rightarrow \log 0.003 = \log 3 - 3 - - - - - - (1)
Using this we can evaluate log⁡3\log 3 or we know the value of log⁡3\log 3.
We know, log⁡3=0.4771\log 3 = 0.4771 (approx)
Now, substituting the value of log⁡3\log 3 in (1), we get
⇒log⁡0.003=0.4771−3\Rightarrow \log 0.003 = 0.4771 - 3
⇒log⁡0.003=−2.5229\Rightarrow \log 0.003 = - 2.5229
Hence, we get
log⁡0.003=−2.5229\log 0.003 = - 2.5229
So, the correct answer is “ - 2.5229”.

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