TITLE: Solve $ \left( \log_3 x \right)^2 + \log_3 (x^2) + 1 = 0$
QUESTION [5 upvotes]: I'm new to logarithms and I am having trouble solving this equation
$$ \left( \log_3 x \right)^2 + \log_3 (x^2) + 1 = 0.$$
How would I solve this? A step-by-step response would be appreciated.
Also, I know how to solve it with assigning $\log_3 (x)$ as $x$ and solving $x^2 + x + 1 = 0$, and getting the answer from there. I am looking to see how I would do it with just logarithms and no quadratics. 
Thanks

REPLY [4 votes]: You should get the answer $x = \frac{1}{3}$. 
Here is why. Solving the quadratic equation $w^2 + 2w + 1=0$ gives $w=-1$ as a repeated root. Substituting $w =\log_3(x)$ in $w = -1$ gives $\log_3(x) = -1$ 
Which implies $3^{\log_3(x)} = 3^{-1} \implies x = \frac{1}{3}$. (we used the inverse function $3^x$ of our function $\log_3 (x)$ ).