Let `f(x)=log_(1//3)((log)_(1/3)((log)_7(sinx+a))` be defined for every real value of `x ,` then the possible value(s) of `a` is (a) 3 (b) 4 (c) 5 (d) 6

Options

A. 3
B. 4
C. 5
D. 6

Explanation

To solve the problem, we need to analyze the function \( f(x) = \log_{1/3}(\log_{1/3}(\log_7(\sin x + a))) \) and determine the possible values of \( a \) such that this function is defined for every real value of \( x \).

### Step 1: Conditions for the logarithmic functions
For the logarithmic functions to be defined, the arguments of all logarithms must be positive. Therefore, we need to ensure:
1. \( \log_7(\sin x + a) > 0 \)
2. \( \log_{1/3}(\log_7(\sin x + a)) > 0 \)

### Step 2: Analyzing the first condition
The first condition \( \log_7(\sin x + a) > 0 \) implies:
\[ \sin x + a > 7^0 = 1 \]
This leads to:
\[ \sin x + a > 1 \implies a > 1 - \sin x \]

### Step 3: Finding the minimum value of \( \sin x \)
The sine function oscillates between -1 and 1. Therefore, the minimum value of \( \sin x \) is -1. Substituting this into the inequality gives:
\[ a > 1 - (-1) \implies a > 2 \]

### Step 4: Analyzing the second condition
Next, we analyze the second condition \( \log_{1/3}(\log_7(\sin x + a)) > 0 \). Since the base \( \frac{1}{3} < 1 \), this implies:
\[ \log_7(\sin x + a) < 1 \]
This leads to:
\[ \sin x + a < 7^1 = 7 \implies a < 7 - \sin x \]

### Step 5: Finding the maximum value of \( \sin x \)
The maximum value of \( \sin x \) is 1. Substituting this into the inequality gives:
\[ a < 7 - 1 \implies a < 6 \]

### Step 6: Combining the inequalities
From the two conditions, we have:
\[ 2 < a < 6 \]

### Step 7: Identifying possible values of \( a \)
Now, we check the given options:
- (a) 3: Valid, since \( 2 < 3 < 6 \)
- (b) 4: Valid, since \( 2 < 4 < 6 \)
- (c) 5: Valid, since \( 2 < 5 < 6 \)
- (d) 6: Invalid, since \( 6 \) is not less than \( 6 \)

### Conclusion
The possible values of \( a \) are:
- \( 3 \)
- \( 4 \)
- \( 5 \)

### Final Answer
The possible values of \( a \) are (a) 3, (b) 4, and (c) 5.