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The natural logarithmic function, denoted as ln(x), is the inverse of the exponential function e^x, where e is Euler's number. It is crucial for understanding continuous growth and decay in fields like mathematics and economics. This function's graph, properties, and its application in calculus, particularly in differentiation and integration, are explored. Additionally, its significance in finance through continuous compounding is discussed, demonstrating its practical applications.

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## Definition and Properties

### Definition

The natural logarithmic function is the inverse of the exponential function and is represented as \( f(x) = \ln(x) \)

### Properties

Graph

The graph of the natural logarithmic function mirrors the exponential function and has a domain of positive real numbers and a range of all real numbers

Identities

The natural logarithm follows the identities \( \ln(e^x) = x \) and \( e^{\ln(x)} = x \), showing the inverse relationship between logarithmic and exponential functions

### Applications

The natural logarithmic function is essential for modeling continuous growth and decay in various disciplines, including mathematics, physics, and economics

## Continuous Compounding

### Definition

Continuous compounding is a method of calculating interest or growth continuously over time

### Calculation

The natural logarithm is used to calculate the time needed for an investment to grow by a specific factor with continuous compounding

### Example

If an investment compounds continuously at a 100% annual interest rate, it would take approximately 3 years to grow 20 times in size, as determined by \( \ln(20) \)

## Transforming Logarithmic Functions

### Change-of-Base Formula

The change-of-base formula, \( \log_b(x) = \frac{\log_a(x)}{\log_a(b)} \), allows for the conversion of logarithmic functions to the natural base, \( e \)

### Comparison

Transforming logarithmic functions to the natural base allows for comparisons between different logarithmic expressions

### Proportionality

Logarithms with various bases are proportional to the natural logarithm by a constant factor

## Calculus Applications

### Differentiation

The derivative of \( \ln(x) \) with respect to \( x \) is \( \frac{d}{dx}\ln(x) = \frac{1}{x} \), an important formula for solving problems related to rates of change

### Integration

The integral of \( \ln(x) \), \( \int \ln(x) \, dx \), yields \( x\ln(x) - x + C \), expanding the utility of the natural logarithmic function in mathematical analysis and problem-solving

### Relevance

The natural logarithmic function is integral to calculus and its applications in real-world scenarios, highlighting its relevance in both educational and practical settings

Algorino

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