Question

If $a,b$ are co-prime numbers and satisfying $(2+\sqrt{3}{)}^{\frac{1}{{\log }_a(2-\sqrt{3})}+\frac{1}{{\log }_b\left(\frac{\sqrt{3}-1}{\sqrt{3}+1}\right)}}=\frac{1}{12}$, then $(a+b)$ can be is equal to

• A. 13
• B. 5
• C. 7
• D. 8

Answer 1

Answer: (C)

As, $\frac{1}{{\log }_a(2-\sqrt{3})}+\frac{1}{{\log }_b\left(\frac{\sqrt{3}-1}{\sqrt{3}+1}\right)}={\log }_{2-\sqrt{3}}a+{\log }_{\frac{\sqrt{3}-1}{\sqrt{3}+1}}b$
$={\log }_{2-\sqrt{3}}a+{\log }_{2-\sqrt{3}}b={\log }_{2-\sqrt{3}}\text{ (ab) }$
Now, $(2+\sqrt{3}{)}^{{\log }_{2-\sqrt{3}}(ab)}=\frac{1}{12}\Rightarrow (2-\sqrt{3}{)}^{{\log }_{2-\sqrt{3}}\left(\frac{1}{ab}\right)}=\frac{1}{12}$
$\Rightarrow \frac{1}{ab}=\frac{1}{12}\Rightarrow ab=12$
As $a,b$ are co-prime numbers, so either $a=4,b=3$ or $a=3,b=4$. Hence, $(a+b)=7$.