Continuity of Logarithmic Functions

The logarithmic function is an important type of function in calculus; understanding its continuity properties 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
• Inverse relation: the logarithmic function and the exponential function are inverses of each other
• Graph features: a smooth curve, with no jumps or breaks

Basic Logarithmic Functions

The Natural Logarithmic Function

Properties:

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

The General Logarithmic Function

Properties:

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

Proof of the Continuity of Logarithmic Functions

Proof Strategy

1. Use the monotonicity of the logarithmic function

   • The logarithmic function is monotone on $(0, +\infty)$

2. Use the properties of limits

   • For every $x_0 \in (0, +\infty)$, $\lim_{x \to x_0} \log_a x = \log_a x_0$

3. Prove continuity at an arbitrary point of the domain

   • Since $\lim_{x \to x_0} f(x) = f(x_0)$
   • Therefore the function is continuous at $x_0$

Composite Logarithmic Functions

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

Analysis:

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

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

Analysis:

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

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

Analysis:

• The inner function $g(x) = \cos x$ is continuous on $\mathbb{R}$
• The outer function $h(x) = \ln x$ is continuous on $(0, +\infty)$
• We need $\cos x > 0$
• Therefore the interval of continuity is all the values for which $\cos x > 0$ holds, i.e. all the $x$ values that satisfy it

Graph Characteristics of Logarithmic Functions

Summary of Graph Features

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

Applications of Logarithmic Functions

1. Scientific Modelling

The logarithmic function is widely used in science:

• pH value: describes the acidity of a solution $\text{pH} = -\log_{10}[\text{H}^+]$
• Earthquake intensity: the Richter magnitude $M = \log_{10} \frac{A}{A_0}$
• Sound intensity: the decibel level $L = 10 \log_{10} \frac{I}{I_0}$

2. Economic Applications

The logarithmic function has important applications in economics:

• Diminishing returns: describes diminishing marginal utility
• Growth rate: describes relative growth rates
• Elasticity analysis: describes the elasticity of demand

3. Engineering Applications

The logarithmic function has important applications in engineering:

• Signal processing: describes signal gain
• Data compression: describes information entropy
• Frequency response: describes the response of a system

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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

Given $f(x) = \ln(2x - 3)$, determine its intervals of continuity.

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Summary

Symbols Used in This Article