What is (log3 x)^2=log3 (x^2) + 3 ?

Since we have the term log3 x^2, we'll use the power property of
logarithms:log3 x^2 = 2 log3 x.

The given equation (log3 x)^2 = log3
x^2 + 3 converts to

(log3 x)^2 - 2log3 x - 3 =
0

We'll substitute  log3 x = u.

We'll
re-write the equation:

 u^2 - 2u -3
=0

Factoring, we'll get:

(u-3)(u+1)
=0

We'll put each factor as zero:

u-3 =
0

So, u = 3

u + 1 =
0

u =  -1

But log3 x = 3 => x =
3^3 => x = 27

log3 x = -1
=> x = 3^-1 => x =
1/3

Since both solutions are positive,
we'll accept them.