Question

If ${\log }_e\frac{a+b}{2}=\frac{{\log }_{e}a+{\log }_{e}b}{2}$ then which one of the following is true

• A. $a=2b$
• B. $a=b$
• C. $a=3b$
• D. none of these

Answer 1

The correct option is B $a=b$

Given,

${\log }_e\frac{a+b}{2}=\frac{{\log }_{e}a+{\log }_{e}b}{2}$

or, ${\log }_e\frac{a+b}{2}=\frac{1}{2}{\log }_{e}ab$

or, ${\log }_e\frac{a+b}{2}={\log }_e\sqrt{ab}$

or, $\frac{a+b}{2}=\sqrt{ab}$

or, $a-2\sqrt{ab}+b=0$

or, $(\sqrt{a}-\sqrt{b}{)}^2=0$

or, $\sqrt{a}=\sqrt{b}$

or, $a=b$.