If x>0, and log2⁡ x+log2⁡ (x)+log2⁡ (x4)+log2⁡ (x8)+log2⁡ (x16)+…=4then x equals

If $x > 0,$ and $\log_{2} x + \log_{2} (\sqrt{x}) +$ $\log_{2} (\sqrt[4]{x}) + \log_{2} (\sqrt[8]{x}) + \log_{2} (\sqrt[16]{x}) + \dots = 4$
then $x$ equals

A
2
B
3
C
4
D
5

Solution:

We can write the given equation as

$\log_{2} (x^{1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots}) = 4$

$\Rightarrow \log_{2} (x^{2}) = 4 \Rightarrow x^{2} = 2^{4} \Rightarrow x = 4$

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