The domain of the function `f(x) = log_10 log_10 (1 + x ^3)` is

Options

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

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 1: Ensure the inner logarithm is positive
The first step is to find the condition under which the inner logarithm \( \log_{10}(1 + x^3) \) is defined and positive.

For \( \log_{10}(1 + x^3) \) to be defined:
1. The argument must be greater than 0:
\[ 1 + x^3 > 0 \]

### Step 2: Solve the inequality
Next, we solve the inequality \( 1 + x^3 > 0 \):
\[ x^3 > -1 \]
This inequality holds true for all \( x > -1 \).

### Step 3: Ensure the outer logarithm is positive
Now, we need to ensure that the outer logarithm \( \log_{10}(y) \) is also positive, where \( y = \log_{10}(1 + x^3) \). For this logarithm to be defined and positive:
\[ \log_{10}(1 + x^3) > 0 \]

### Step 4: Solve the inequality for the outer logarithm
This implies:
\[ 1 + x^3 > 10^0 \]
\[ 1 + x^3 > 1 \]
\[ x^3 > 0 \]
This means \( x > 0 \).

### Step 5: Combine the conditions
From the above steps, we have two conditions:
1. \( x > -1 \)
2. \( x > 0 \)

The stricter condition is \( x > 0 \). Therefore, the domain of the function \( f(x) \) is:
\[ (0, \infty) \]

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

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