Question

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

• A. $\frac{a}{a+b}$
• B. $\frac{a}{1+b}$
• C. $\frac{ab}{a+b}$
• D. $\frac{abc}{a+b}$

Answer 1

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