`(6)/(5)a^(log_(a)xlog_(10)alog_(a)5)-3^(log_(10)(x//10))=9^(log_(100)x+log_(4)2)`, then x=

Options

A. `10^(0)`
B. `10^(1)`
C. `10^(2)`
D. none

Answer

To solve the equation

\[ \frac{6}{5} a^{\log_a x \log_{10} a \log_a 5} - 3^{\log_{10} \left(\frac{x}{10}\right)} = 9^{\log_{100} x + \log_4 2} \]

we will break it down step by step.

### Step 1: Simplifying the Left-Hand Side

1. **Convert the logarithm expressions**:
- We know that \(\log_a x = \frac{\log_{10} x}{\log_{10} a}\).
- Therefore, we can rewrite \(a^{\log_a x}\) as \(x\).

So, we have:

\[ \frac{6}{5} a^{\left(\frac{\log_{10} x}{\log_{10} a}\right) \log_{10} a \log_a 5} = \frac{6}{5} a^{\log_{10} x \cdot \frac{\log_{10} 5}{\log_{10} a}} = \frac{6}{5} x^{\log_a 5} \]

2. **Simplifying \(3^{\log_{10} \left(\frac{x}{10}\right)}\)**:
- This can be rewritten as \(3^{\log_{10} x - 1} = \frac{3^{\log_{10} x}}{3}\).

So, the left-hand side becomes:

\[ \frac{6}{5} x^{\log_a 5} - \frac{3^{\log_{10} x}}{3} \]

### Step 2: Simplifying the Right-Hand Side

1. **Convert \(9^{\log_{100} x + \log_4 2}\)**:
- We know that \(9 = 3^2\), so we can rewrite it as:

\[ (3^2)^{\log_{100} x + \log_4 2} = 3^{2(\log_{100} x + \log_4 2)} \]

2. **Simplifying \(\log_{100} x\)**:
- \(\log_{100} x = \frac{\log_{10} x}{\log_{10} 100} = \frac{\log_{10} x}{2}\).
- \(\log_4 2 = \frac{1}{2}\).

So, we have:

\[ 3^{2\left(\frac{\log_{10} x}{2} + \frac{1}{2}\right)} = 3^{\log_{10} x + 1} = 3^{\log_{10} (10x)} = 10x \]

### Step 3: Equating Both Sides

Now we equate the left-hand side and the right-hand side:

\[ \frac{6}{5} x^{\log_a 5} - \frac{3^{\log_{10} x}}{3} = 10x \]

### Step 4: Solving for \(x\)

1. **Rearranging the equation**:
- Move all terms to one side:

\[ \frac{6}{5} x^{\log_a 5} - 10x - \frac{3^{\log_{10} x}}{3} = 0 \]

2. **Substituting values**:
- This equation may require numerical methods or specific values for \(a\) to solve for \(x\).

### Final Solution

The solution for \(x\) will depend on the values of \(a\) and may require numerical approximation methods to find the exact value.