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What is ${{\log }_{10}}\left( \dfrac{3}{2} \right)+{{\log }_{10}}\left( \dfrac{4}{3} \right)+{{\log }_{10}}\left( \dfrac{5}{4} \right)+....$ up to 8 terms equal to?
(a) 0
(b) 1
(c) ${{\log }_{10}}5$
(d) None of the above

Last updated date: 20th Jul 2024
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Hint: Assume the given expression as E and write all the 8 terms of logarithm. Now, use the formula of log given as $\log \left( m \right)+\log \left( n \right)=\log \left( m\times n \right)$ for all the 8 terms. Simplify the argument of the resultant log expression by cancelling the common factors in the multiplication. The remaining argument with its log to base 10 will be our answer.

Complete step by step solution:
Here we have been provided with the expression ${{\log }_{10}}\left( \dfrac{3}{2} \right)+{{\log }_{10}}\left( \dfrac{4}{3} \right)+{{\log }_{10}}\left( \dfrac{5}{4} \right)+....$ and we have to find its value up to 8 terms. Let us assume the given expression as E, so we have,
\[\Rightarrow E={{\log }_{10}}\left( \dfrac{3}{2} \right)+{{\log }_{10}}\left( \dfrac{4}{3} \right)+{{\log }_{10}}\left( \dfrac{5}{4} \right)+....\] up to 8 terms
On observing the above pattern of the arguments of log expressions we can write the above expression with its last term as: -
\[\Rightarrow E={{\log }_{10}}\left( \dfrac{3}{2} \right)+{{\log }_{10}}\left( \dfrac{4}{3} \right)+{{\log }_{10}}\left( \dfrac{5}{4} \right)+.....+{{\log }_{10}}\left( \dfrac{10}{9} \right)\]
Using the formula of log given as $\log \left( m \right)+\log \left( n \right)=\log \left( m\times n \right)$ for all the 8 terms we get,
\[\Rightarrow E={{\log }_{10}}\left( \dfrac{3\times 4\times 5\times 6\times 7\times 8\times 9\times 10}{2\times 3\times 4\times 5\times 6\times 7\times 8\times 9} \right)\]
Now, cancelling the common factors to simplify the argument of the above log expression we get,
  & \Rightarrow E={{\log }_{10}}\left( \dfrac{10}{2} \right) \\
 & \therefore E={{\log }_{10}}\left( 5 \right) \\

So, the correct answer is “Option c”.

Note: Note that you can solve the above question using different formulas of log. In one of the approaches you can use the formula given as $\log \left( \dfrac{m}{n} \right)=\log \left( m \right)-\log \left( n \right)$ to simplify each term. Further you need to cancel the log expressions having common arguments to get the expression ${{\log }_{10}}\left( 10 \right)-{{\log }_{10}}\left( 2 \right)$. Finally, use the formula $\log \left( m \right)-\log \left( n \right)=\log \left( \dfrac{m}{n} \right)$ and simplify the argument to get the answer. Remember all the formulas of log functions in which one important formula is $\log \left( {{a}^{m}} \right)=m\log \left( a \right)$.