Question

The value of \(\rm log_3(\sqrt{27\sqrt{27\sqrt{27}}}......)\) is 
1. 9
2. 3
3. 27
4. 1

Answer 1

Correct Answer - Option 2 : 3

Concept:

\(\rm log\ x^n=nlog \ x\)

\(\rm log_aa=1\)

 

Calculation:

Let, \(\rm x =\sqrt{27\sqrt{27\sqrt{27}}}......\)

On squaring both the sides, we get 

\(\rm x^2 ={27\sqrt{27\sqrt{27}}}......\)

\(⇒ \rm x^2 ={27x}\)              (∵ \(\rm x =\sqrt{27\sqrt{27\sqrt{27}}}......\))

\(⇒ \rm x^2 -{27x}=0\)

⇒ x(x - 27) = 0

⇒ x = 27 OR x = 0

x can't be zero.

So, x = 27

\(\Rightarrow \rm \sqrt{27\sqrt{27\sqrt{27}}}......=27\)

Now, \(\rm log_3(\sqrt{27\sqrt{27\sqrt{27}}}......) =log_327\)

\(\rm =log_33^3\)

\(\rm =3log_33\)                   (∵ \(\rm log\ x^n=nlog \ x\))

= 3                            (∵ \(\rm log_aa=1\))

Hence, option (2) is correct.