If `a gt 0 , 2 log_(x) a + log_(ax) a + 3 log_(a^(2)x) a = 0 ` then x =

Options

A. `a^(1//2)`
B. `a^(-1//2)`
C. `a^(-2//3)`
D. `a^(-4//3)`

Answer

To solve the equation \( 2 \log_{x} a + \log_{ax} a + 3 \log_{a^2 x} a = 0 \), we will follow these steps:

### Step 1: Rewrite the logarithmic expressions
Using the change of base formula, we can rewrite the logarithms:
\[ \log_{x} a = \frac{\log a}{\log x}, \quad \log_{ax} a = \frac{\log a}{\log(ax)} = \frac{\log a}{\log a + \log x}, \quad \log_{a^2 x} a = \frac{\log a}{\log(a^2 x)} = \frac{\log a}{2\log a + \log x} \]

### Step 2: Substitute the rewritten logarithms into the equation
Substituting these into the original equation gives:
\[ 2 \cdot \frac{\log a}{\log x} + \frac{\log a}{\log a + \log x} + 3 \cdot \frac{\log a}{2\log a + \log x} = 0 \]

### Step 3: Factor out \(\log a\)
Since \(\log a\) cannot be zero (as \(a > 0\)), we can factor it out:
\[ \log a \left( 2 \cdot \frac{1}{\log x} + \frac{1}{\log a + \log x} + 3 \cdot \frac{1}{2\log a + \log x} \right) = 0 \]

### Step 4: Set the remaining expression to zero
We need to solve:
\[ 2 \cdot \frac{1}{\log x} + \frac{1}{\log a + \log x} + 3 \cdot \frac{1}{2\log a + \log x} = 0 \]

### Step 5: Clear the denominators
Multiply through by \(\log x (\log a + \log x)(2\log a + \log x)\) to eliminate the fractions:
\[ 2(\log a + \log x)(2\log a + \log x) + \log x(2\log a + \log x) + 3\log x(\log a + \log x) = 0 \]

### Step 6: Expand and simplify
Expanding each term:
\[ 2(2\log a^2 + 3\log a \log x + \log^2 x) + (\log x(2\log a + \log x)) + 3(\log x \log a + \log^2 x) = 0 \]
Combine like terms.

### Step 7: Solve the resulting polynomial equation
This will yield a quadratic equation in terms of \(\log x\). Let \(t = \log x\):
\[ At^2 + Bt + C = 0 \]
where \(A\), \(B\), and \(C\) are coefficients derived from the previous step.

### Step 8: Use the quadratic formula
Using the quadratic formula:
\[ t = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]

### Step 9: Substitute back to find \(x\)
Convert back from \(t\) to \(x\):
\[ x = 10^t \]

### Final Step: Calculate the values of \(x\)
Calculate the two possible values of \(x\) based on the two roots obtained from the quadratic formula.