Find x if x^(1+log3 x)=9x^2 .

We'll take logarithms both
sides:

log3 [x^(1+log3 x)] = log3
(9x^2)

We'll apply the power rule to the left
side:

(1+log3 x)*(log3 x) = log3
(9x^2)

We'll apply the product rule to the right
side:

(1+log3 x)*(log3 x) = log3 9 + log3
(x^2)

We'll remove the
brackets:

log3 x + (log3 x)^2 = 2 + 2log3
x

We'll move all terms to one
side:

(log3 x)^2 + log3 x - 2log3 x - 2 =
0

We'll combine like
terms:

(log3 x)^2 - log3 x - 2 =
0

We'll note log3 x = t

t^2 -
t - 2 = 0

We'll apply quadratic
formula:

t1 = [1 + sqrt(1 +
8)]/2

t1 = (1 + 3)/2

t1 =
2

t2 = -1

log3 x =
t1

log3 x = 2

x =
3^2

x1 = 9

log3 x =
t2

log3 x = -1

x2 =
3^-1

x2 =
1/3

The solutions are: {1/3 ;
9}.