`(6)/(5)a^((log_(a)x)(log_(10)a)(log_(a)5))-3^(log_(10)((x)/(10)))=9^(log_(100)x+log_(4)2)`
(where `a gt 0, a ne 1)`, then `log_(3)x=alpha +beta, alpha ` is integer, `beta in [0, 1)`, then `alpha=`

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 follow these steps:

### Step 1: Simplify the Right-Hand Side

The right-hand side of the equation is

\[ 9^{\log_{100} x + \log_4 2}. \]

Using the property of logarithms, we can rewrite this as:

\[ 9^{\log_{100} x} \cdot 9^{\log_4 2}. \]

Now, we know that \(9 = 3^2\), so we can rewrite it as:

\[ (3^2)^{\log_{100} x} \cdot 9^{\log_4 2} = 3^{2 \log_{100} x} \cdot 9^{\log_4 2}. \]

Next, we can simplify \(9^{\log_4 2}\):

\[ 9^{\log_4 2} = (3^2)^{\log_4 2} = 3^{2 \log_4 2} = 3^{\log_4 4} = 3^1 = 3. \]

Thus, the right-hand side becomes:

\[ 3^{2 \log_{100} x} \cdot 3 = 3^{2 \log_{100} x + 1}. \]

### Step 2: Simplify the Left-Hand Side

Now, let's simplify the left-hand side:

\[ \frac{6}{5} a^{(\log_a x)(\log_{10} a)(\log_a 5)} - 3^{\log_{10} \left(\frac{x}{10}\right)}. \]

Using the properties of logarithms, we can express \(\log_{10} \left(\frac{x}{10}\right)\) as:

\[ \log_{10} x - \log_{10} 10 = \log_{10} x - 1. \]

Thus,

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

Now, substituting this back into the left-hand side gives us:

\[ \frac{6}{5} a^{(\log_a x)(\log_{10} a)(\log_a 5)} - \frac{3^{\log_{10} x}}{3}. \]

### Step 3: Equate Both Sides

Now we equate both sides:

\[ \frac{6}{5} a^{(\log_a x)(\log_{10} a)(\log_a 5)} - \frac{3^{\log_{10} x}}{3} = 3^{2 \log_{100} x + 1}. \]

### Step 4: Let \(t = \log_{10} x\)

Let \(t = \log_{10} x\). Then we have:

\[ \frac{6}{5} a^{(\log_a 10^t)(\log_{10} a)(\log_a 5)} - \frac{3^t}{3} = 3^{2 \frac{t}{2} + 1}. \]

### Step 5: Solve for \(x\)

After simplifying and solving for \(x\), we find:

\[ x = 100. \]

### Step 6: Find \(\log_3 x\)

Now, we need to find \(\log_3 x\):

\[ \log_3 100 = \log_3 (10^2) = 2 \log_3 10. \]

### Step 7: Determine \(\alpha\) and \(\beta\)

We know that:

\[ \log_3 10 \approx 2.0959, \]

so

\[ \log_3 100 \approx 2 \times 2.0959 = 4.1918. \]

Thus, we can express this as:

\[ \log_3 100 = 4 + 0.1918, \]

where \(\alpha = 4\) and \(\beta \approx 0.1918\).

### Final Answer

Therefore, the value of \(\alpha\) is:

\[ \boxed{4}. \]