The domain of the function `log_(10) log_(10)` (1 + x^3) `is

Options

A. `(0,oo)`
B. `[o,oo)`
C. `(-1,oo)`
D. `(-1,0)`

Answer

To find the domain of the function \( f(x) = \log_{10}(\log_{10}(1 + x^3)) \), we need to ensure that the arguments of both logarithmic functions are positive.

### Step-by-Step Solution:

1. **Identify the Inner Function**:
The inner function is \( \log_{10}(1 + x^3) \). For this logarithm to be defined, the argument must be greater than zero:
\[ 1 + x^3 > 0 \]

2. **Solve the Inequality**:
Rearranging the inequality gives:
\[ x^3 > -1 \]
Taking the cube root of both sides, we find:
\[ x > -1 \]
Thus, the first condition for the domain is:
\[ x \in (-1, \infty) \]

3. **Identify the Outer Function**:
Now we need to ensure that the output of the inner function \( \log_{10}(1 + x^3) \) is also positive:
\[ \log_{10}(1 + x^3) > 0 \]

4. **Solve the Second Inequality**:
For the logarithm to be positive, its argument must be greater than 1:
\[ 1 + x^3 > 1 \]
Simplifying this gives:
\[ x^3 > 0 \]
Taking the cube root again, we find:
\[ x > 0 \]
Thus, the second condition for the domain is:
\[ x \in (0, \infty) \]

5. **Combine the Conditions**:
The domain of the function \( f(x) \) is the intersection of the two conditions:
\[ (-1, \infty) \cap (0, \infty) = (0, \infty) \]

### Final Answer:
The domain of the function \( f(x) = \log_{10}(\log_{10}(1 + x^3)) \) is:
\[ \boxed{(0, \infty)} \]