Solve for x : x^(1+log 2 sqrtx) = 16

This is a logarithmic
equation.

We'll take logarithms both side of the equation
and the logarithms will have the base 2:

log 2 [x^(1+log 2
sqrtx)] = log 2 16

We'll use the power property of
logarithms and we'll write 16 = 2^4:

(1+log 2 sqrtx) * log
2 x= log 2 2^4

[1+log 2 (x)^1/2] * log 2 x= 4*log 2
2

{1 + [log 2 (x)]/2}* log 2 x=
4

We'll remove the brackets form the left
side:

log 2 x + [(log 2 x)^2]/2 - 4 =
0

2*log 2 x + (log 2 x)^2 - 4 =
0

We'll substitute log 2 x =
t

t^2 + 2t - 4 = 0

We'll apply
the quadratic formula:

t1 =
[-2+sqrt(4+16)]/2

t1 =
(-2+2sqrt5)/2

t1 = -1+sqrt5

t2
= -1-sqrt5

 log 2 x =
t1

x1 =
2^-1+sqrt5

x2 =
2^-(1+sqrt5)

x2 =
1/2^(1+sqrt5)