If `f(x)=logx^(4) and g(x)=4logx` then the the domain for which f(x) and g(x) are identical ?

Options

A. `(-oo,oo)`
B. `[0,oo)`
C. `(0,oo)`
D. none of these

Answer

To determine the domain for which the functions \( f(x) = \log(x^4) \) and \( g(x) = 4\log(x) \) are identical, we can follow these steps:

### Step 1: Determine the domain of \( f(x) \)

The function \( f(x) = \log(x^4) \) is defined when the argument of the logarithm is greater than zero. Since \( x^4 \) is always positive for \( x > 0 \), we have:

\[ x^4 > 0 \implies x > 0 \]

Thus, the domain of \( f(x) \) is:

\[ \text{Domain of } f(x) = (0, \infty) \]

### Step 2: Determine the domain of \( g(x) \)

The function \( g(x) = 4\log(x) \) is also defined when the argument of the logarithm is greater than zero. Therefore, we have:

\[ x > 0 \]

Thus, the domain of \( g(x) \) is:

\[ \text{Domain of } g(x) = (0, \infty) \]

### Step 3: Compare the domains of \( f(x) \) and \( g(x) \)

Now that we have both domains, we can compare them:

\[ \text{Domain of } f(x) = (0, \infty) \]
\[ \text{Domain of } g(x) = (0, \infty) \]

Since both domains are identical, we conclude that:

\[ \text{Domain of } f(x) \text{ and } g(x) \text{ are identical.} \]

### Conclusion

The domain for which \( f(x) \) and \( g(x) \) are identical is:

\[ (0, \infty) \]

### Final Answer

The required domain is \( (0, \infty) \).

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