Question

How many distinct real numbers belong to the following collection?
 $ \left\{ {\left. {\ln \left( {2 - \sqrt 3 } \right),\ln \left( {2 + \sqrt 3 } \right), - \ln \left( {2 - \sqrt 3 } \right), - \ln \left( {2 + \sqrt 3 } \right),\ln \left( {\dfrac{{2 + \sqrt 3 }}{{2 - \sqrt 3 }}} \right),\ln (7 + 4\sqrt 3 )} \right\}} \right. $
(A) $ 2 $
(B) $ 3 $
(C) $ 4 $
(D) $ 5 $

Answer 1

Hint: In the question a collection of terms is given and we are asked to find the number of distinct real numbers belonging to the following collection, so take each of the terms separately and simplify it, then repeat the same procedure for all the given terms. Then only, you can find the number of distinct real numbers in collection.

Complete step-by-step answer:
In this question it is given that
 $ \left\{ {\left. {\ln \left( {2 - \sqrt 3 } \right),\ln \left( {2 + \sqrt 3 } \right), - \ln \left( {2 - \sqrt 3 } \right), - \ln \left( {2 + \sqrt 3 } \right),\ln \left( {\dfrac{{2 + \sqrt 3 }}{{2 - \sqrt 3 }}} \right),\ln (7 + 4\sqrt 3 )} \right\}} \right. $
We have to find the number of distinct real roots.
So, Case 1:
 $ \ln \left( {2 - \sqrt 3 } \right) $ . . . . (given)
On multiplying and dividing with $ \ln \left( {2 + \sqrt 3 } \right) $ we get,
 $ \ln \left( {2 - \sqrt 3 } \right) = {\ln \left( {2 - \sqrt 3 } \right) \times \dfrac{{2 + \sqrt 3 }}{{2 + \sqrt 3 }}} $
On simplifying the above equation, we get,
 $ \ln \left( {2 - \sqrt 3 } \right) = \ln {\left( {2 + \sqrt 3 } \right)^{ - 1}} $
By using $ \log \left( {\ln } \right) $ properties we get,
 $ \ln \left( {2 - \sqrt 3 } \right) = - \ln \left( {2 + \sqrt 3 } \right) $
Case 2:

 $ \ln \left( {2 + \sqrt 3 } \right) $ . . . . (given)
We will use the same way which we have used in case (1)
 $ \ln \left( {2 + \sqrt 3 } \right) $
On multiplying and dividing with $ \left( {2 + \sqrt 3 } \right) $ in the above equation we get,
 $ \ln \left( {2 + \sqrt 3 } \right) = - \ln \left( {2 - \sqrt 3 } \right) $
Case 3:
 $ - \ln \left( {2 - \sqrt 3 } \right) $
We will use the same way which we have used in both the above cases.
 $ - \ln \left( {2 - \sqrt 3 } \right) $
On multiplying and dividing with $ \left( {2 + \sqrt 3 } \right) $ in above equation we get,
 $ - \ln \left( {2 - \sqrt 3 } \right) = \ln \left( {2 + \sqrt 3 } \right) $
Case 4:
 $ \ln \left( {2 + \sqrt 3 } \right) $
On multiplying and dividing with $ \left( {2 - \sqrt 3 } \right) $ in the above equation we get ,
 $ - \ln \left( {2 + \sqrt 3 } \right) = - \log {\left( {2 - \sqrt 3 } \right)^{ - 1}} $
By applying log rule method in above equation we get,
 $ \ln \left( {2 + \sqrt 3 } \right) = \ln \left( {2 - \sqrt 3 } \right) $
Case 5:
 $ \ln \left( {\dfrac{{2 + \sqrt 3 }}{{2 - \sqrt 3 }}} \right) $
By multiplying and dividing with $ \left( {2 + \sqrt 3 } \right) $ in above equation we get,
 $ \ln \left( {\dfrac{{2 + \sqrt 3 }}{{2 - \sqrt 3 }}} \right) = \ln {\left( {2 + \sqrt 3 } \right)^2} $
By using $ \log $ method we get,
 $ \ln \left( {\dfrac{{2 + \sqrt 3 }}{{2 - \sqrt 3 }}} \right) = 2\ln \left( {2 + \sqrt 3 } \right) $
Case 6:
 $ \ln (7 + 4\sqrt 3 ) $
 $ \ln (7 + 4\sqrt 3 ) $ can also be written as $ \ln \left( {4 + 3 + \sqrt 3 } \right) $
Also, $ \ln \left( {7 + 4\sqrt 3 } \right) = \ln {\left( {2 + \sqrt 3 } \right)^2} $
On applying $ \log $ rule in above equation
We get $ \ln (7 + 4\sqrt 3 ) = 2\ln \left( {2 + \sqrt 3 } \right) $
Now we write all the solved terms from the above.
 $ - \ln \left( {2 + \sqrt 3 } \right) $ , $ - \ln \left( {2 - \sqrt 3 } \right) $ , $ \ln \left( {2 + \sqrt 3 } \right) $ , $ \ln \left( {2 - \sqrt 3 } \right) $ , $ \ln \left( {2 + \sqrt 3 } \right) $
There are three common terms. So, the number of distinct value $ = 2 $
Therefore, from the above explanation the correct option is [A] $ 2 $ .

Note: In this type of question it is better to use basic log concepts such as , $ \ln {\left( k \right)^{ - 1}} = - \ln \left( k \right) $ , where $ k \in R $ (i.e. k belongs to any real number).