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

Question

Given x is a real number, If SQRT(x+15) + SQRT(x)=15 then find the value of x.

Answer 1

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.

Comments

May need a minor correction as its a infinite series, do you like to correct it :)

---

Thanks, I've done the needful.

Answer 2

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