Question

The following exercises are intended to derive the fundamental properties of the natural log starting from the definition ln(x)=?x1dtt, using properties of the definite integral and making no further assumptions. Use the identity \(\ln x=\int_{1}^{x} \frac{d t}{x}\) to show that \(\ln (x)\) is an increasing function of \(x\) on \([0, \infty)\), and use the previous exercises to show that the range of \(\ln (x)\) is \((-\infty, \infty) .\) Without any further assumptions, conclude that \(\ln (x)\) has an inverse function defined on \((-\infty, \infty)\).

Short answer

\( \ln(x) \) is increasing on \((0, \infty)\) with range \((-\infty, \infty)\), hence it has an inverse: the exponential function.

Step-by-step solution

Identify the problem context

We need to determine if \( \ln(x) \) is an increasing function on \([0, \infty)\) and find the range of \( \ln(x) \) using the definition \( \ln(x)=\int_{1}^{x} \frac{dt}{t} \). Additionally, we need to show that \( \ln(x) \) has an inverse function on \(( -\infty, \infty)\).

Apply the First Fundamental Theorem of Calculus

The First Fundamental Theorem of Calculus states that if \( F(x) = \int_{a}^{x} f(t) \, dt \), then \( F'(x) = f(x) \). Given \( \ln(x) = \int_{1}^{x} \frac{dt}{t} \), it follows that \( \frac{d}{dx}\ln(x) = \frac{1}{x} \).

Determine the sign of the derivative

The derivative \( \frac{1}{x} \) is positive for all \( x > 0 \). Therefore, \( \ln(x) \) is an increasing function on \((0, \infty)\).

Explore the behavior of \( \ln(x) \) as \( x \) approaches the boundaries

For \( x < 1 \), \( \ln(x) = \int_{1}^{x} \frac{dt}{t} = -\int_{x}^{1} \frac{dt}{t} \) which is a negative area, and \( \ln(x) \rightarrow -\infty \) as \( x \rightarrow 0^{+} \). For \( x > 1 \), \( \ln(x) \rightarrow \infty \) as \( x \rightarrow \infty \). Hence, the range of \( \ln(x) \) is \((-\infty, \infty)\).

Conclude about the inverse function based on monotonicity and range

Since \( \ln(x) \) is strictly increasing and its range covers all real numbers, it has an inverse function. The inverse will map all real numbers \(( -\infty, \infty) \) to \( (0, \infty) \). The inverse function is the exponential function.

Key concepts

Understanding the Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a special kind of logarithm that has the base \( e \), where \( e \approx 2.718 \ldots \). The natural logarithm is used frequently in mathematics and the sciences because it simplifies many mathematical expressions.
One distinctive feature of the natural logarithm is that it's defined as an integral:

• \( \ln(x) = \int_{1}^{x} \frac{1}{t} \, dt \)

This definition provides a way to describe the natural log based on the area under the curve of \( \frac{1}{t} \) from 1 to \( x \). The natural logarithm is defined for all positive real numbers \( x > 0 \).
Understanding how \( \ln(x) \) behaves can lead us to important conclusions about its properties and its relation to other functions, like the exponential function, which serves as its inverse.

First Fundamental Theorem of Calculus

The First Fundamental Theorem of Calculus connects differentiation with integration, forming a bridge between these two central concepts in calculus. Here's what it states in simple terms:

• If \( F(x) = \int_{a}^{x} f(t) \, dt \), then the derivative \( F'(x) = f(x) \).

This is powerful because it allows us to find how a quantity changes (differentiationenousyd by understanding the area accumulated so far (integrationenous).
In the context of the natural logarithm, we used this theorem to show that:

• \( \frac{d}{dx} \ln(x) = \frac{1}{x} \)

Because \( \frac{1}{x} \) is positive for all \( x > 0 \), it tells us that \( \ln(x) \) is an increasing function. This is a key property because it leads us to the conclusion that the function is one-to-one, which is necessary for it to have an inverse.

Understanding Inverse Functions

Inverse functions are a fascinating topic in mathematics, offering a way to reverse the operation of the original function. For a function to have an inverse:

• It must be one-to-one (strictly increasing or decreasing).
• It must cover the whole range of possible outputs.

Since we've established that \( \ln(x) \) is strictly increasing over \( (0, \infty) \) and its range is \( (-\infty, \infty) \), it qualifies to have an inverse. This inverse function is the exponential function, \( e^y \), which maps back from \( (-\infty, \infty) \) to \( (0, \infty) \).
In simpler terms, if you take the natural logarithm of a number, using its inverse function, the exponential, will return the original number. For instance:

• If \( y = \ln(x) \), then \( x = e^y \).

Understanding how these inverse operations work is crucial, not only in solving logarithmic and exponential equations but also in fields as diverse as engineering, physics, and finance.