Exponential and Logarithmic Functions of Natural Base, e

Introduction to the Natural Base e

The natural base, denoted e, is a fundamental mathematical constant approximately equal to 2.71828. It arises naturally in problems involving continuous growth or decay.

• Definition:

• Continuous Exponential Growth: If a quantity P grows continuously at an annual percentage rate r (as a decimal), its value after t years is .

Example: Continuous Growth

• Luis invests A = 5000e^{0.0325 \times 5} \approx 5882.24$.

Natural Logarithms

The natural logarithm of a positive number x is the logarithm to the base e, denoted as ln x. It is the inverse function of the exponential function.

• Definition: if and only if

Properties of Natural Logarithms

• for all real numbers

• for all

Example: Evaluating Natural Logarithms

Domain of Logarithmic Functions

The natural logarithm function is defined only for . Its domain is and its range is .

• For , solve to find the domain.

Example: Domain

• For , domain is .

Solving Exponential and Logarithmic Equations

• To solve , take the natural logarithm of both sides:

• To solve , exponentiate both sides:

Example: Solving Exponential Equation

Exponential Growth Rate and Doubling Time

The exponential growth rate r and the doubling time T are related by:

Example: Doubling Time

• If a population doubles every 8 months, (8.7% per month).

Derivatives of Exponential (Base-e) Functions

Differentiation Rules

• The derivative of is .

• For , (Chain Rule).

Examples

Applications: Growth of an Account

• If , then gives the instantaneous rate of growth.

Example

• ,

Derivatives of Natural Logarithmic Functions

Differentiation Rules

• for

• For ,

Examples

Application: Forgetting Curve in Psychology

The percentage of students retaining information over time can be modeled by a logarithmic function.

• Example: ,

• At ,

• Rate of change at : (percentage decreases by 13.5% per minute at $t = 2$)

Applications: Uninhibited and Limited Growth Models

Uninhibited (Exponential) Growth

• Differential equation:

• Solution: , where is the initial population

Logistic Growth Model

• Differential equation: , where is the carrying capacity

• Solution:

Example: Netflix Revenue Model

• models revenue in billions of dollars, years after 2016

• Point of diminishing returns: billion dollars

Applications: Exponential Decay

Exponential Decay Model

• Differential equation:

• Solution:

Half-Life and Decay Rate

• Relationship: , ,

Example: Carbon Dating

• Half-life of carbon-14: 5730 years

• Decay rate:

• If 22.3% of carbon-14 is lost, age years

Present Value with Continuous Compounding

• Present value: , where is the future value, is the interest rate, is time

Newton’s Law of Cooling

• Differential equation: , where is the ambient temperature

• Solution:

Example: Forensics – Time of Death

• Given two temperature readings and room temperature, solve for and to estimate time of death.

Derivatives of and

Exponential Functions with Base

Logarithmic Functions with Base

Examples

Base-n Exponential Growth

• If a quantity increases n-fold every units of time:

Example: Population Decline

• If a population loses 15% every 4 years: