**Solution:**

To find the value of f(x), we need to integrate the absolute value of the given function from 0 to x.

Let's break down the given function step by step.

**Step 1:**
f(x) = ∫[0 to x] |log₂(log₃(log₄(cos(t * a))))| dt

**Step 2:**
Let's evaluate the innermost logarithmic expression first.

log₄(cos(t * a))

**Step 3:**
Next, we take the logarithm base 3 of the above expression.

log₃(log₄(cos(t * a)))

**Step 4:**
Finally, we take the logarithm base 2 of the above expression.

log₂(log₃(log₄(cos(t * a))))

**Step 5:**
Now, we need to find the absolute value of the above expression.

|log₂(log₃(log₄(cos(t * a))))|

**Step 6:**
We integrate the absolute value of the function from 0 to x.

f(x) = ∫[0 to x] |log₂(log₃(log₄(cos(t * a))))| dt

**Step 7:**
To evaluate the integral, we need to break it into different intervals based on the behavior of the function.

**Case 1:** When the expression inside the absolute value is positive.
In this case, we can directly integrate the expression from 0 to x.

**Case 2:** When the expression inside the absolute value is negative.
In this case, we need to change the sign of the expression and integrate it from 0 to x.

**Step 8:**
After integrating the expression in each interval, we need to add the absolute values of the results to get the final value of f(x).

**Conclusion:**
The value of f(x) depends on the behavior of the expression inside the absolute value function. By evaluating the integral in different intervals, we can find the value of f(x) for a given value of x.