Exponential and Logarithmic Functions

Introduction

Exponential and logarithmic functions are fundamental in Business Calculus, especially for modeling growth, decay, and various economic phenomena. This section focuses on the differentiation of natural logarithmic functions and their applications.

Derivatives of Natural Logarithmic Functions

Objectives

• Differentiate functions involving natural logarithms.

• Solve applied problems involving natural logarithmic functions and their derivatives.

Theorem 7: Derivative of the Natural Logarithm

The derivative of the natural logarithm function is a foundational result in calculus.

• Statement: For any positive number x:

• Key Point: The rate of change of the natural logarithm function decreases as x increases.

Example 1: Differentiation Practice

• a) Differentiate :

• b) Differentiate :

• c) Differentiate :

Theorem 8: Derivative of the Logarithm of a Function

• Generalization: The derivative of the natural logarithm of a function uses the chain rule.

• Key Point: This formula is essential for differentiating composite logarithmic functions.

Example 2: Differentiating Composite Logarithmic Functions

• a) Differentiate :

• b) Differentiate :

Quick Check 2: Practice Problems

• a) Differentiate :

• b) Differentiate :

• c) Differentiate :

Applications: Social Science - Forgetting Curve

Logarithmic Learning Model

Logarithmic functions are used to model phenomena such as memory retention over time. In a psychological experiment, the percentage of students retaining syllables after t minutes is modeled by:

,   for

• Key Point: The function decreases as time increases, representing the forgetting process.

Example: Calculating Retention and Rate of Change

• Find the percentage retained at :

• Find the rate of change at :

• Interpretation: Two minutes after the experiment, the percentage of students retaining the syllables is decreasing at a rate of 13.5% per minute.

Example: When Does Retention Drop to 20%?

• Set and solve for :

• Conclusion: The percentage of students who retained the syllables drops to 20% after about 9.23 minutes.

Graphical Representation

The graph of shows a decreasing curve, illustrating the rapid initial decline in retention, which slows over time.

Summary Table: Logarithmic Differentiation Rules

Additional info: Logarithmic differentiation is especially useful for functions involving products, quotients, or powers, and is widely applied in business, economics, and social sciences for modeling rates of change and growth/decay processes.