Sure! Let's solve the equation step-by-step:
Given the equation:
[tex]\[ 3 \log_9 x = (\log_3 x)^2 \][/tex]
First, we need to express [tex]\(\log_9 x\)[/tex] in terms of base 3 logs. We know that:
[tex]\[ \log_9 x = \frac{\log_3 x}{\log_3 9} \][/tex]
Since [tex]\(9 = 3^2\)[/tex], we have:
[tex]\[ \log_3 9 = 2 \][/tex]
Therefore:
[tex]\[ \log_9 x = \frac{\log_3 x}{2} \][/tex]
Now we substitute this back into the original equation:
[tex]\[ 3 \left( \frac{\log_3 x}{2} \right) = (\log_3 x)^2 \][/tex]
Simplify the equation:
[tex]\[ \frac{3}{2} \log_3 x = (\log_3 x)^2 \][/tex]
Let's introduce a new variable to make it easier to solve. Let:
[tex]\[ y = \log_3 x \][/tex]
The equation becomes:
[tex]\[ \frac{3}{2} y = y^2 \][/tex]
Rearrange the equation to standard quadratic form:
[tex]\[ y^2 - \frac{3}{2} y = 0 \][/tex]
Factor out [tex]\(y\)[/tex] from the equation:
[tex]\[ y \left( y - \frac{3}{2} \right) = 0 \][/tex]
This gives us two potential solutions for [tex]\(y\)[/tex]:
[tex]\[ y = 0 \quad \text{or} \quad y = \frac{3}{2} \][/tex]
Now we need to convert back from [tex]\(y\)[/tex] to [tex]\(\log_3 x\)[/tex]:
1. If [tex]\(y = 0\)[/tex]:
[tex]\[ \log_3 x = 0 \][/tex]
This implies:
[tex]\[ x = 3^0 = 1 \][/tex]
2. If [tex]\(y = \frac{3}{2}\)[/tex]:
[tex]\[ \log_3 x = \frac{3}{2} \][/tex]
This implies:
[tex]\[ x = 3^{\frac{3}{2}} = \sqrt{27} = 3\sqrt{3} \][/tex]
Thus, the solutions to the equation [tex]\(3 \log_9 x = (\log_3 x)^2\)[/tex] are:
[tex]\[ x = 1 \quad \text{and} \quad x = 3\sqrt{3} \][/tex]
So, the solutions are [tex]\(x = 1\)[/tex] and [tex]\(x = 3\sqrt{3}\)[/tex].
