Question

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

Answer 1

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 \[2\log 9 - \log 18\]
We know that \[2\log a = \log {a^2}\]
By using this formula, we get
\[ \Rightarrow 2\log 9 - \log 18 = \log {9^2} - \log 18\]
We know that \[\log a - \log b = \log \left( {\dfrac{a}{b}} \right)\]
By using this formula, we get
\[
   \Rightarrow 2\log 9 - \log 18 = \log {9^2} - \log 18 = \log \left( {\dfrac{{{9^2}}}{{18}}} \right) \\
   \Rightarrow 2\log 9 - \log 18 = \log \left( {\dfrac{{9 \times 9}}{{18}}} \right) \\
   \Rightarrow 2\log 9 - \log 18 = \log \left( {\dfrac{9}{2}} \right) \\
  \therefore 2\log 9 - \log 18 = \log 4.5 \\
\]
The value of \[2\log 9 - \log 18\] is \[\log 4.5\]
Thus, the correct option is C. \[\log 4.5\]

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