The number of integral values of `a` for which `f(x)="log"((log)_(1/3)((log)_7(sinx+a)))` is defined for every real value of `x` is ________

Explanation

To determine the number of integral values of \( a \) for which the function

\[ f(x) = \log_{\frac{1}{3}} \left( \log_{7} (\sin x + a) \right) \]

is defined for every real value of \( x \), we need to ensure that the argument of each logarithm is valid.

### Step-by-Step Solution:

1. **Understanding the Logarithmic Function**:
The function \( f(x) \) is defined if:
- \( \log_{7} (\sin x + a) > 0 \)
- \( \log_{7} (\sin x + a) < 1 \)

2. **Setting Up the Inequalities**:
From the first condition \( \log_{7} (\sin x + a) > 0 \):
\[ \sin x + a > 1 \quad \Rightarrow \quad a > 1 - \sin x \]

From the second condition \( \log_{7} (\sin x + a) < 1 \):
\[ \sin x + a < 7 \quad \Rightarrow \quad a < 7 - \sin x \]

3. **Combining the Inequalities**:
We can combine these inequalities:
\[ 1 - \sin x < a < 7 - \sin x \]

4. **Finding the Range of \( \sin x \)**:
The range of \( \sin x \) is from -1 to 1. Therefore, we will evaluate the inequalities at the extreme values of \( \sin x \).

- When \( \sin x = -1 \):
\[ 1 - (-1) < a < 7 - (-1) \quad \Rightarrow \quad 2 < a < 8 \]

- When \( \sin x = 1 \):
\[ 1 - 1 < a < 7 - 1 \quad \Rightarrow \quad 0 < a < 6 \]

5. **Finding the Intersection of the Ranges**:
Now we need to find the intersection of the two ranges:
- From \( \sin x = -1 \): \( 2 < a < 8 \)
- From \( \sin x = 1 \): \( 0 < a < 6 \)

The intersection of these two ranges is:
\[ 2 < a < 6 \]

6. **Determining Integral Values**:
The integral values of \( a \) that satisfy \( 2 < a < 6 \) are:
- \( a = 3, 4, 5 \)

7. **Counting the Integral Values**:
Therefore, the number of integral values of \( a \) is \( 3 \).

### Final Answer:
The number of integral values of \( a \) for which \( f(x) \) is defined for every real value of \( x \) is \( \boxed{3} \).