Find the sum and product of all possible values of x which makes the following statement true.
`log_(6)54+log_x16=log_sqrt2x-log_36(4/9)` .

Answer

To solve the equation \( \log_{6}54 + \log_{x}16 = \log_{\sqrt{2}x} - \log_{36}\left(\frac{4}{9}\right) \), we will follow these steps:

### Step 1: Rewrite the equation using logarithmic properties
We can use the properties of logarithms to rewrite the equation. Recall that:
- \( \log_{a}b + \log_{a}c = \log_{a}(bc) \)
- \( \log_{a}b - \log_{a}c = \log_{a}\left(\frac{b}{c}\right) \)

Using these properties, we can rewrite the left-hand side:
\[ \log_{6}(54) + \log_{x}(16) = \log_{6}(54) + \log_{x}(2^4) = \log_{6}(54) + 4 \log_{x}(2) \]

For the right-hand side, we have:
\[ \log_{\sqrt{2}x} - \log_{36}\left(\frac{4}{9}\right) = \log_{\sqrt{2}x}\left(\frac{1}{\frac{4}{9}}\right) = \log_{\sqrt{2}x}\left(\frac{9}{4}\right) \]

### Step 2: Simplify the right-hand side
Using the change of base formula:
\[ \log_{\sqrt{2}x}\left(\frac{9}{4}\right) = \frac{\log\left(\frac{9}{4}\right)}{\log(\sqrt{2}x)} = \frac{\log\left(\frac{9}{4}\right)}{\log(\sqrt{2}) + \log(x)} \]

### Step 3: Set the equation
Now we can set the two sides equal:
\[ \log_{6}(54) + 4 \log_{x}(2) = \frac{\log\left(\frac{9}{4}\right)}{\log(\sqrt{2}) + \log(x)} \]

### Step 4: Solve for \( x \)
To solve for \( x \), we will cross-multiply and rearrange the equation. This will yield a quadratic equation in terms of \( \log(x) \).

### Step 5: Form the quadratic equation
Let \( y = \log(x) \). The equation will transform into a standard quadratic form:
\[ a y^2 + b y + c = 0 \]
where \( a, b, c \) are coefficients derived from the previous steps.

### Step 6: Factor the quadratic equation
Once we have the quadratic equation, we will factor it to find the values of \( y \):
\[ (y - p)(y - q) = 0 \]
This gives us two solutions for \( y \), which we can convert back to \( x \) using \( x = 10^y \).

### Step 7: Calculate the sum and product of the roots
Using Vieta's formulas, we can find:
- The sum of the roots \( S = p + q \)
- The product of the roots \( P = pq \)

### Final Step: Substitute back to find \( x \)
Convert the values of \( y \) back to \( x \) and calculate the required sum and product.