91Ó°ÊÓ

The natural logarithm function, denoted as \( \ln(x) \), is a fundamental mathematical function. It is the inverse of the exponential function \( e^x \), which is characterized by a base that is an irrational constant approximately equal to 2.71828. The natural logarithm function is only defined for positive numbers, meaning that \( x \) must be greater than zero for \( \ln(x) \) to exist.
\[ \ln(x) \text{ is defined for } x > 0 \]
Because the function is the inverse of an exponential, it "un-does" what exponentiation does. For example, since \( e^y = x \), then \( \ln(x) = y \). This important property allows us to solve equations involving exponential growth or decay.
When dealing with a more complex logarithmic argument, like \( \ln(x - x^2) \), you need to ensure that the entire argument \( (x - x^2) \) remains positive. This forms the basis for finding the domain of the function.

In calculus, critical points of a function are where the function's derivative is zero or undefined. However, in this context, critical points are used to find where a function changes behavior, especially in solving inequalities.
To find the critical points of an expression like \( x(1 - x) \), set the expression equal to zero:
\[ x(1 - x) = 0 \]
This can be solved using the Zero Product Property, which tells us that if \( ab = 0 \), then either \( a = 0 \) or \( b = 0 \). Here, our solutions are:

• \( x = 0 \)
• \( x = 1 \)

These points indicate where the product \( x(1 - x) \) is zero and potentially changes sign. Understanding the sign changes is crucial for determining intervals where the inequality \( x(1-x) > 0 \) holds.

Inequalities are used to determine the set of values that satisfy certain conditions. When solving \( x(1 - x) > 0 \), the idea is to find intervals where this statement is true.
Firstly, determine critical points where \( x(1 - x) = 0 \), which are \( x = 0 \) and \( x = 1 \). These critical points divide the number line into three intervals:

• \((-\infty, 0)\)
• \((0, 1)\)
• \((1, \infty)\)

To find the intervals where \( x(1 - x) > 0 \), you select a test point from each interval and check:

• For \( x = -0.5 \) in \((-\infty, 0)\), \( x(1-x) < 0 \)
• For \( x = 0.5 \) in \((0, 1)\), \( x(1-x) > 0 \)
• For \( x = 1.5 \) in \((1, \infty)\), \( x(1-x) < 0 \)

Through this testing, you determine that the inequality holds true only within the interval \( (0, 1) \). Hence, the domain of the function \( g(x) = \ln(x - x^2) \) is \( (0, 1) \). Solving inequalities like this helps define domains where functions such as logarithms are valid.