Question

If $2^a=3^b=6^c$ then

• A. $c=\frac{ab}{a+b}$
• B. $c=\frac{a+b}{a-b}$
• C. $c=\frac{a{b}^2}{a^2+b^2}$
• D. $c=\frac{a{b}^2}{a^2-b^2}$

Answer 1

The correct option is A $c=\frac{ab}{a+b}$

Let $2^a=3^b=6^c=k$
So, $2^a=k$
$\Rightarrow k^{\frac{1}{a}}=2$ .....(i)
Similarly $k^{\frac{1}{b}}=3$ .....(ii)
$k^{\frac{1}{c}}=6$ .....(iii)
Now, we know that $6=2\times 3$
Putting the values of 6, 2 and 3 from (i), (ii) and (iii) we get:
$k^{\frac{1}{c}}=k^{\frac{1}{a}}\times k^{\frac{1}{b}}\Rightarrow k^{\frac{1}{c}}=k^{\frac{1}{a}+\frac{1}{b}}\Rightarrow \frac{1}{c}=\frac{1}{a}+\frac{1}{b}\Rightarrow \frac{1}{c}=\frac{b+a}{ab}\Rightarrow c=\frac{ab}{a+b}$