The values of x, satisfying the equation for `AA a > 0, 2log_(x) a + log_(ax) a +3log_(a^2 x)a=0` are

Options

A. one
B. two
C. three
D. four

Answer

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

### Step 1: Rewrite the logarithmic expressions
Using the change of base formula, we can rewrite the logarithmic terms:

1. \( \log_a x = \frac{1}{\log_x a} \)
2. \( \log_{ax} a = \log_a a + \log_a x = 1 + \log_a x \)
3. \( \log_{a^2 x} a = \log_a a + \log_a (a^2 x) = 1 + 2 \log_a a + \log_a x = 1 + 2 + \log_a x = 3 + \log_a x \)

Thus, we can rewrite the equation as:
\[ 2 \log_a x + (1 + \log_a x) + 3(3 + \log_a x) = 0 \]

### Step 2: Combine the terms
Now, we can combine the terms:
\[ 2 \log_a x + 1 + \log_a x + 9 + 3 \log_a x = 0 \]
Combine like terms:
\[ (2 + 1 + 3) \log_a x + (1 + 9) = 0 \]
This simplifies to:
\[ 6 \log_a x + 10 = 0 \]

### Step 3: Isolate \( \log_a x \)
Now, isolate \( \log_a x \):
\[ 6 \log_a x = -10 \]
\[ \log_a x = -\frac{10}{6} = -\frac{5}{3} \]

### Step 4: Convert back to exponential form
Convert the logarithmic equation back to exponential form:
\[ x = a^{-\frac{5}{3}} = \frac{1}{a^{\frac{5}{3}}} \]

### Step 5: Conclusion
Thus, the value of \( x \) satisfying the equation is:
\[ x = \frac{1}{a^{\frac{5}{3}}} \]

### Final Answer
The values of \( x \) satisfying the equation are \( \frac{1}{a^{\frac{5}{3}}} \).

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