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Question: The value of \[2\log 9 - \log 18\] is equal to A. \[\log 9\] B. \[ - \log 9\] C. \[\log 4.5\]...

The value of 2log⁡9−log⁡182\log 9 - \log 18 is equal to
A. log⁡9\log 9
B. −log⁡9 - \log 9
C. log⁡4.5\log 4.5
D. −log⁡4.5 - \log 4.5

Explanation

Solution

Hint:First of all, convert the given expression by using logarithm product rule. Then simplify the obtained expression by using logarithm quotient rule to obtain the required answer. So, use this concept to reach the solution of the given problem.

Complete step-by-step answer:
Given expression is 2log⁡9−log⁡182\log 9 - \log 18
We know that 2log⁡a=log⁡a22\log a = \log {a^2}
By using this formula, we get
⇒2log⁡9−log⁡18=log⁡92−log⁡18\Rightarrow 2\log 9 - \log 18 = \log {9^2} - \log 18
We know that log⁡a−log⁡b=log⁡(ab)\log a - \log b = \log \left( {\dfrac{a}{b}} \right)
By using this formula, we get

⇒2log⁡9−log⁡18=log⁡92−log⁡18=log⁡(9218) ⇒2log⁡9−log⁡18=log⁡(9×918) ⇒2log⁡9−log⁡18=log⁡(92) ∴2log⁡9−log⁡18=log⁡4.5  \Rightarrow 2\log 9 - \log 18 = \log {9^2} - \log 18 = \log \left( {\dfrac{{{9^2}}}{{18}}} \right) \\\ \Rightarrow 2\log 9 - \log 18 = \log \left( {\dfrac{{9 \times 9}}{{18}}} \right) \\\ \Rightarrow 2\log 9 - \log 18 = \log \left( {\dfrac{9}{2}} \right) \\\ \therefore 2\log 9 - \log 18 = \log 4.5 \\\

The value of 2log⁡9−log⁡182\log 9 - \log 18 is log⁡4.5\log 4.5
Thus, the correct option is C. log⁡4.5\log 4.5

Note: Here we have used the formulae, logarithm power rule i.e., 2log⁡a=log⁡a22\log a = \log {a^2} and logarithm quotient rule log⁡a−log⁡b=log⁡(ab)\log a - \log b = \log \left( {\dfrac{a}{b}} \right). In mathematics, the base value of log is ee whose value is approximately equal to 2.718.