It is known that x=9 is root of the equation.`log_lamda(x^2+15a^2)-log_lamda(a-2)=log_lamda "(8ax)/(a-2)` find the other roots of this equation.

Answer

To solve the equation given in the problem, we will follow the steps outlined in the video transcript to find the other roots of the equation.

### Step-by-Step Solution:

1. **Write the given equation:**
\[ \log_\lambda(x^2 + 15a^2) - \log_\lambda(a - 2) = \log_\lambda\left(\frac{8ax}{a - 2}\right) \]

2. **Apply the logarithmic property:**
Using the property of logarithms that states \(\log_a(b) - \log_a(c) = \log_a\left(\frac{b}{c}\right)\), we can rewrite the left side:
\[ \log_\lambda\left(\frac{x^2 + 15a^2}{a - 2}\right) = \log_\lambda\left(\frac{8ax}{a - 2}\right) \]

3. **Eliminate the logarithm:**
Since the logarithm is equal on both sides, we can set the arguments equal to each other:
\[ \frac{x^2 + 15a^2}{a - 2} = \frac{8ax}{a - 2} \]
Assuming \(a - 2 \neq 0\), we can multiply both sides by \(a - 2\):
\[ x^2 + 15a^2 = 8ax \]

4. **Rearrange the equation:**
Rearranging gives us:
\[ x^2 - 8ax + 15a^2 = 0 \]

5. **Factor the quadratic equation:**
We can factor the quadratic:
\[ (x - 3a)(x - 5a) = 0 \]
Therefore, the roots are:
\[ x = 3a \quad \text{and} \quad x = 5a \]

6. **Use the known root to find \(a\):**
We know that one of the roots is \(x = 9\). We can set up two equations:
- From \(x = 3a\):
\[ 9 = 3a \implies a = 3 \]
- From \(x = 5a\):
\[ 9 = 5a \implies a = \frac{9}{5} \]

7. **Check the conditions for \(a\):**
Since \(a\) must be greater than 2, we have:
- \(a = 3\) is valid.
- \(a = \frac{9}{5} = 1.8\) is not valid.

8. **Find the other root:**
Substitute \(a = 3\) back into the roots:
- For \(x = 3a\):
\[ x = 3 \times 3 = 9 \quad \text{(given root)} \]
- For \(x = 5a\):
\[ x = 5 \times 3 = 15 \]

Thus, the other root of the equation is:
\[ \boxed{15} \]