Question

Find the domain of each function. $$ g(x)=\frac{1}{\ln x} $$

Short answer

The domain is \( (0, 1) \cup (1, \infty) \).

Step-by-step solution

Identify the restriction from the denominator

Since the function has a denominator, the denominator cannot be zero. Hence, we need to find when \(\ln x = 0\). This occurs when \(x = 1\). Therefore, \(x\) cannot be \(1\).

Evaluate the domain of the logarithmic function

The natural logarithm function \(\ln x\) is only defined for \(x > 0\). Thus, \(x\) must be positive.

Combine the restrictions

Since \(x\) must be positive and cannot be \(1\), the domain of the function is \(x > 0\) and \(x eq 1\).

Key concepts

Logarithmic Functions

Logarithmic functions are fundamental concepts in mathematics, especially in calculus and algebra. A logarithmic function is an inverse of an exponential function. The general form is written as \(\text{log}_b(x)\) where \(b\) is the base. In the context of our exercise, we work with the natural logarithm, which has a base of \(e\) (approximately 2.71828).

The natural logarithm function is denoted as \(\text{ln}(x)\). Logarithmic functions are useful for solving equations involving exponentials by transforming multiplicative relationships into additive ones.

• The natural logarithm of a positive number \(\text{ln}(x)\) is defined only when x > 0.
• The result of \(\text{ln}(1)\) is 0 because \(e^0 = 1\).
• The function \(\text{ln}(x)\) increases slowly and becomes larger as x increases.

Function Restrictions

In mathematics, functions often come with certain restrictions that limit their domain (the set of input values). For example, the function \(\text{g(x) = \frac{1}{\text{ln}(x)}}\) given in the exercise has specific restrictions because not all x-values are permissible.

• First, the natural logarithm function \(\text{ln}(x)\) is defined only for \(\text{x} > 0\). So, x must be positive for \(\text{g(x)}\) to exist.
• Second, \(\text{g(x)}\) has a denominator \(\text{ln}(x)\), and a function's denominator cannot be zero, as division by zero is undefined.
• Thus, we require that \(\text{ln}(x) ≠ 0\), which implies that x ≠ 1, because \(\text{ln}(1) = 0\).

By combining these restrictions, we determine that the domain of g(x) is \(\text{x > 0 and x ≠ 1. }\)

Natural Logarithm

The natural logarithm \(\text{ln}(x)\) is a special type of logarithmic function where the base is the irrational number \(e\) (Euler's number). Euler's number is approximately 2.71828 and has special properties that make the natural logarithm widely used in mathematics and science.

The function is written as \(\text{y = \text{ln}(x)}\) and serves as the inverse of the exponential function \(\text{y = e^x}\).

• Properties of the natural logarithm include:
• The natural logarithm of 1 is zero, \(\text{ln}(1) = 0\), because \(\text{e}^0 = 1\).
• The function \(\text{ln}(x)\) is continuous for all positive x-values and is undefined for negative values and zero.
• It is always increasing, meaning that as x increases, \(\text{ln}(x)\) also increases.
• The natural logarithm has real numbers as its range, which means it can output any real number.
< \br>It is important to clearly understand these properties to accurately work with natural logarithms in various mathematical problems.