Problem 63 - Calculus

Question

Give an example of: A function \(f(x)\) such that \(\ln (f(x))\) is only defined for \(x<0\).

Step-by-Step Solution

Answer

A function satisfying the condition is \(f(x) = -x\).

Step 1: Understanding Logarithmic Function Domain

The natural logarithmic function \(\ln(x)\) is only defined for \(x > 0\). Therefore, for \(\ln(f(x))\) to be defined, \(f(x)\) must be greater than zero.

Step 2: Determine Conditions for f(x)

To ensure that \(\ln(f(x))\) is only defined for \(x < 0\), \(f(x)\) should be positive only when \(x < 0\). For \(x \geq 0\), \(f(x)\) should be negative or zero so that \(\ln(f(x))\) remains undefined.

Step 3: Choose a Suitable Function

A simple function that is only positive for \(x < 0\) is \(f(x) = -x\). This function is positive when \(x < 0\) because the negative of a negative number is positive.

Step 4: Verify the Function

For \(x < 0\): \(f(x) = -x > 0\), satisfying the condition for \(\ln(f(x))\) to be defined.For \(x \geq 0\): \(f(x) = -x \leq 0\), thus \(\ln(f(x))\) is undefined. Hence, \(f(x) = -x\) is the correct function.

Key Concepts

Function Domain

In mathematics, the **domain** of a function refers to the set of input values for which the function is defined. When discussing logarithmic functions, understanding the domain is crucial. The natural logarithm function, denoted as \(\ln(x)\), is only defined when \(x > 0\). This means any input to a natural logarithm must be positive, as the logarithm of zero or a negative number is not defined in the set of real numbers.

• Natural logarithms can only handle positive numbers.
• The domain determines where the function is valid and operational.

For logarithmic functions involving a composite function like \(\ln(f(x))\), the domain is influenced by both the natural logarithm's domain and the range of \(f(x)\). To keep \(\ln(f(x))\) within a valid range for calculation, \(f(x)\) itself must satisfy the condition of being positive.

Natural Logarithm

The **natural logarithm**, denoted \(\ln(x)\), is a logarithm to the base \(e\), where \(e\) is an irrational constant approximately equal to 2.71828. It is a significant function in mathematics due to its properties and applicability to many natural processes.

• \(\ln(x)\) is useful for analyzing exponential growth or decay.
• It converts multiplicative processes into additive processes, simplifying complex calculations.

When we consider a function \(\ln(f(x))\), the value of \(f(x)\) must be greater than zero. Therefore, the domain of \(\ln(f(x))\) depends on where \(f(x)\) remains positive. Understanding the properties of the natural logarithm helps us manipulate and transform various problems, especially when paired with other functions.

Conditions for Positivity

When working with logarithmic functions, specifically \(\ln(f(x))\), ensuring the **positivity condition** is essential. This condition requires that the function \(f(x)\) remains positive to make the logarithm defined. If we want \(\ln(f(x))\) to be defined only for \(x < 0\), the focus should be on crafting a function that fulfills this particular condition.

• To meet the condition, \(f(x)\) must be positive when \(x < 0\).
• For \(x \geq 0\), \(f(x)\) should result in a non-positive value to keep \(\ln(f(x))\) undefined.

One such function is \(f(x) = -x\). This is because:

• When \(x < 0\), \(-x > 0\), fulfilling the positive requirement.
• When \(x \geq 0\), \(-x \leq 0\), ensuring \(\ln(f(x))\) becomes undefined.

By carefully selecting and understanding the behavioral circumstances of a function, we can control the domain of a logarithmic expression effectively.