Question

what is the range of ( y = log_{e} x )?

• all real numbers less than 0
• all real numbers greater than 0
• all real numbers not equal to 0
• all real numbers

Explanation:

Step1: Recall the definition of logarithmic function

The logarithmic function \( y = \log_{a}x \) (where \( a>0, a
eq1 \)) is the inverse of the exponential function \( y = a^{x} \).

Step2: Analyze the range of exponential function

The range of the exponential function \( y = a^{x} \) ( \( a>0, a
eq1 \)) is \( (0, +\infty) \).

Step3: Determine the range of logarithmic function

Since the logarithmic function \( y=\log_{a}x \) is the inverse of the exponential function \( y = a^{x} \), the domain of the logarithmic function is the range of the exponential function, and the range of the logarithmic function is the domain of the exponential function. The domain of the exponential function \( y = a^{x} \) is all real numbers. So the range of \( y=\log_{a}x \) ( \( a > 0,a
eq1 \)) is all real numbers.

Answer:

all real numbers (the option: all real numbers)