Logarithmic Function

The logarithmic function is an important type of function in calculus with distinctive properties. Understanding the continuity properties of logarithmic functions is essential for studying more complicated types of function.

Basic Properties

The logarithmic function has the following basic properties:

• Monotonicity: the logarithmic function is a monotone function
• Continuity: continuous on its domain
• Differentiability: differentiable on its domain
• Inverse relation: the logarithmic function and the exponential function are inverses of each other

The Natural Logarithmic Function

Properties:

• Domain: $(0, +\infty)$
• Continuous on its domain
• The graph is a monotonically increasing logarithmic curve
• Range: $\mathbb{R}$
• Derivative: $f'(x) = \frac{1}{x}$

The General Logarithmic Function

Properties:

• Domain: $(0, +\infty)$
• Continuous on its domain
• Monotonically increasing when $a > 1$, monotonically decreasing when $0 < a < 1$
• Range: $\mathbb{R}$
• Derivative: $f'(x) = \frac{1}{x \ln a}$

Graph Characteristics of Logarithmic Functions

1. Monotonicity: the logarithmic function is monotone on its domain
2. Asymptotes: there is a vertical asymptote $x = 0$
3. Smoothness: the graph is a smooth curve, with no corners or breaks
4. Growth: the logarithmic function grows slowly

Continuity Analysis of Logarithmic Functions

Determining Continuity

The logarithmic function is continuous on $(0, +\infty)$.

Proof of Continuity

The continuity of the logarithmic function rests on the following facts:

1. Continuity of inverse functions: the inverse of a continuous function is continuous
2. Continuity of the exponential function: the exponential function $e^x$ is continuous on $\mathbb{R}$
3. The logarithmic function is the inverse of the exponential function: $\ln x$ is the inverse of $e^x$
4. Hence the logarithmic function is continuous: $\ln x$ is continuous on $(0, +\infty)$

Continuity of Composite Functions

Basic Theorem

If the function $f(x)$ is continuous at $x_0$ and $f(x_0) > 0$, then $\ln f(x)$ is continuous at $x_0$.

Examples

Example 1: $f(x) = \ln(x^2 + 1)$

Analysis:

• The inner function $h(x) = x^2 + 1$ is continuous on $\mathbb{R}$
• The outer function $g(x) = \ln x$ is continuous on $(0, +\infty)$
• Since $x^2 + 1 > 0$ holds for every $x \in \mathbb{R}$
• Therefore $f(x) = g(h(x))$ is continuous on $\mathbb{R}$

Example 2: $f(x) = \ln(x^2 - 1)$

Analysis:

• The inner function $h(x) = x^2 - 1$ is continuous on $\mathbb{R}$
• The outer function $g(x) = \ln x$ is continuous on $(0, +\infty)$
• We need $x^2 - 1 > 0$, that is, $x < -1$ or $x > 1$
• Therefore $f(x)$ is continuous on $(-\infty, -1) \cup (1, +\infty)$

Example 3: $f(x) = \ln(\cos x)$

Analysis:

• The inner function $h(x) = \cos x$ is continuous on $\mathbb{R}$
• The outer function $g(x) = \ln x$ is continuous on $(0, +\infty)$
• We need $\cos x > 0$
• Therefore $f(x)$ is continuous at every value for which $\cos x > 0$ holds, that is, every $x$ value

The Relation Between Logarithmic and Exponential Functions

They Are Inverses of Each Other

The logarithmic function and the exponential function are inverses of each other:

• $\ln(e^x) = x$ for every $x \in \mathbb{R}$
• $e^{\ln x} = x$ for every $x > 0$

Composite Functions

Example: $f(x) = e^{\ln x}$

Analysis:

• The inner function $h(x) = \ln x$ is continuous on $(0, +\infty)$
• The outer function $g(x) = e^x$ is continuous on $\mathbb{R}$
• Therefore $f(x) = g(h(x))$ is continuous on $(0, +\infty)$
• In fact $f(x) = x$ (when $x > 0$)

Exercises

Exercise 1

Determine the continuity of $f(x) = \ln(x^2 + 1)$ on $\mathbb{R}$.

Exercise 2

Determine the intervals of continuity of $f(x) = \ln(x^2 - 1)$.

Exercise 3

Determine the intervals of continuity of $f(x) = \ln(\cos x)$.

Exercise 4

Determine the continuity of $f(x) = \log_2 x$ on $(0, +\infty)$.

---

Summary

Symbols Used in This Article