Question 59509
<pre><font size = 5 color = "indigo"><b>
(log<sub>3</sub>x)<sup>2</sup> + log<sub>3</sub>x<sup>2</sup> + 1 = 0

On the second term use the rule

log<sub>b</sub>a<sup>n</sup> = n·log<sub>b</sub>a

(log<sub>3</sub>x)2 + 2·log<sub>3</sub>x + 1 = 0

Let the letter w, or any letter other
than x, be such that

w = log<sub>3</sub>x

Then the above equation becomes

        (w)<sup>2</sup> + 2·w + 1 = 0
           w<sup>2</sup> + 2w + 1 = 0   
Factoring
        (w + 1)(w + 1) = 0

So the solutions w = -1 and w = -1 are 
equal.

Now since w = log<sub>3</sub>x, then w = -1 becomes

         log<sub>3</sub>x = -1

Now use the rule of logarithms that says:

log<sub>b</sub>a = c can be rewritten as a = b<sup>c</sup> 

to rewrite log<sub>3</sub>x = -1 as

           x = 3<sup>-1</sup>

or         x = 1/3

Edwin</pre>