To determine the domain of the function [tex]\(F(x) = \ln(x)\)[/tex], we need to consider the properties of the natural logarithm function.
1. The natural logarithm function, [tex]\(\ln(x)\)[/tex], is defined only for positive real numbers. This is because the logarithm function represents the inverse of the exponential function, [tex]\(e^x\)[/tex], which is always positive for any real number input.
2. Therefore, for [tex]\(\ln(x)\)[/tex] to be valid, [tex]\(x\)[/tex] must be greater than 0. In other words, [tex]\(\ln(x)\)[/tex] is undefined for [tex]\(x \leq 0\)[/tex].
Given this information, we can identify the correct option for the domain of [tex]\(F(x) = \ln(x)\)[/tex] from the choices given:
A. all real numbers except 0 - This is incorrect because [tex]\(\ln(x)\)[/tex] is not defined for negative numbers.
B. all real numbers greater than 0 - This is correct because the natural logarithm function is defined for any positive real number.
C. all real numbers less than 0 - This is incorrect because [tex]\(\ln(x)\)[/tex] is not defined for any negative numbers or zero as mentioned previously.
D. all real numbers - This is incorrect because [tex]\(\ln(x)\)[/tex] is only defined for positive real numbers, not for zero or negative numbers.
Therefore, the domain of the function [tex]\(F(x) = \ln(x)\)[/tex] is:
B. all real numbers greater than 0
