x = 4.

S(infinite) = a/(1-r)

r = log_2 sqrtx/log_2 x

= 1/2

(log_2 x)/(1-1/2) = 4

log_2 x = 2

log_2 x = log_2 2^2

x = 4.

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What would be the value of x if ---- log_2 x + log_2 sqrt(x) + log_2 sqrt(sqrt(x)) + log_2 sqrt(sqrt(sqrt(x))) .....= 4

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Lets use this property -

log_2 x^n = n log_2 x

So log_2 x + log_2 sqrt(x) + log_2 sqrt(sqrt(x)) + log_2 sqrt(sqrt(sqrt(x))) .....= 4

=> log_2 x + 1/2 log_2 x + 1/4 log_2 x + 1/8 log_2 x .....= 4

=> log_2 x [1 + 1/2 + 1/4 + 1/8 ....] = 4

=> log_2 x [1/(1-1/2)] = 4

=> 2 log_2 x = 4

=> log_2 x = 2

=> x = 2^2

=> x = 4

Answer: 4

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