Question
7. If \( f(x)=\log 2^{x^{2}+4 x-5} \) then, what is the domain of \( \mathrm{f}(\mathrm{x}) \) ?
Ask by Hart Bernard. in Ethiopia
Jan 20,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The domain of \( f(x) \) is all real numbers.
Solution
To determine the domain of the function \( f(x) = \log \left( 2^{x^2 + 4x - 5} \right) \), we need to ensure that the argument of the logarithm is positive because the logarithm is only defined for positive real numbers.
Let's analyze the argument inside the logarithm:
\[
2^{x^2 + 4x - 5}
\]
Since the base \(2\) is a positive number and an exponential function with a positive base is always positive regardless of the exponent, the expression \(2^{x^2 + 4x - 5}\) is always positive for all real values of \(x\).
Therefore, the logarithm function is defined for all real numbers \(x\) in this case.
**Domain of \( f(x) \):** All real numbers \(x\).
**Final Answer:**
All real numbers are allowed. The domain of f is all real numbers.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To determine the domain of \( f(x) = \log 2^{x^{2}+4x-5} \), we need to consider the argument of the logarithm. The logarithm function is defined only for positive values. Therefore, we must have: \[ 2^{x^2 + 4x - 5} > 0 \] Since \( 2^y > 0 \) for any real number \( y \), we only need to ensure that the exponent \( x^2 + 4x - 5 \) is defined. In this case, the function is defined for all real numbers, so the domain of \( f(x) \) is all real numbers, or: \[ \text{Domain of } f(x): (-\infty, \infty) \]
