Let a, b, c be real numbers greater than 1, and n be a positive real number not equal to 1. If $$\log_{n}(\log_{2}a) = 1, \log_{n} (log_{2}b) = 2$$ and $$\log_{n}(\log_{2}c) = 3$$, then which of the following is true?

Given,

$$\log_{n}(\log_{2}a) = 1$$

So, $$\log_2a\ =n^1= n$$

or, $$a=2^n$$

So, $$a^n=\left(2^n\right)^n=2^{n^2}$$

Now, $$\log_{n}(\log_{2}b) = 2$$

So, $$\log_2b\ =n^2$$

or, $$b=2^{n^2}$$

Now, $$a^n+b=2^{n^2}+2^{n^2}=2^{n^2}\times\ 2=2^{n^2+1}$$

So, $$\left(a^n+b\right)^n=2^{\left(n^2+1\right)\cdot n}=2^{n^3+n}$$ ------->(1)

Now, $$\log_{n}(\log_{2}c) = 3$$

So, $$\log_2c\ =n^3$$

or, $$c=2^{n^3}$$

From equation (1), $$\left(a^n+b\right)^n=2^{n^3+n}=2^{n^3}\times\ 2^n=c\times\ a=ac$$

So, option B is the correct answer.