Problem 386 - Calculus Volume 1

Question

The following exercises are intended to derive the fundamental properties of the natural log starting from the definition \(\ln (x)=\int_{1}^{x} \frac{d t}{t}, \quad\) using properties of the definite integral and making no further assumptions. Pretend, for the moment, that we do not know that \(e^{x}\) is the inverse function of \(\ln (x),\) but keep in mind that \(\ln (x)\) has an inverse function defined on \((-\infty, \infty) .\) Call it \(E\) . Use the identity \(\ln x y=\ln x+\ln y\) to deduce that \(E(a+b)=E(a) E(b)\) for any real numbers \(a, b.\)

Step-by-Step Solution

Answer

Using the identity \( \ln(xy) = \ln(x) + \ln(y) \), we derive \( E(a+b) = E(a)E(b) \).

Step 1: Understand the Given Definition and Identity

We are given the definition of the natural logarithm as \( \ln(x) = \int_{1}^{x} \frac{dt}{t} \). We also know from properties of logarithms that \( \ln(xy) = \ln(x) + \ln(y) \). We need to use this to show that if \( E \) is the inverse of \( \ln \), then \( E(a + b) = E(a)E(b) \).

Step 2: Apply the Logarithm Property

Start with the identity \( \ln(xy) = \ln(x) + \ln(y) \). If we apply \( E \) (the inverse function of \( \ln \)) to both sides, we have \( E(\ln(xy)) = E(\ln(x) + \ln(y)) \).

Step 3: Understanding the Inverse Relationship

Since \( E \) is the inverse function of \( \ln \), then \( E(\ln(x)) = x \) and \( E(\ln(y)) = y \). Thus, applying \( E \) to \( \ln(xy) \) gives \( xy \) and applying \( E \) to \( \ln(x) + \ln(y) \) relates to the multiplication of \( E(\ln(x)) \) and \( E(\ln(y)) \).

Step 4: Deduce the Key Relationship

As \( E(\ln(x)) = x \) and \( E(\ln(y)) = y \), applying \( E \) to \( \ln(x) + \ln(y) \) gives us \( E(\ln(x) + \ln(y)) = E(a+b) \) for \( x = E(a) \) and \( y = E(b) \). Thus, \( xy = E(a)E(b) \), which matches the left side \( E(a+b) = xy \).

Step 5: Conclusion

The natural conclusion is that for the identity to hold using the inverse function \( E \), we must have \( E(a + b) = E(a)E(b) \) for any real numbers \( a \) and \( b \). This is consistent with the properties of exponential functions.

Key Concepts

Inverse Functions

Inverse functions are a pair of functions that reverse each other's operations. If you have a function, say \( f(x) \), its inverse \( g(x) \) is such that when \( f \) processes a value and then \( g \) processes the result, you end up back with the original value. In mathematical terms, if \( f(g(x)) = x \) and \( g(f(x)) = x \), then \( g(x) \) is the inverse of \( f(x) \).

Consider the natural logarithm \( \ln(x) \), whose inverse is an exponential function. The inverse of \( \ln(x) \) is often called \( e^x \) because applying exponential functions to logarithms returns the original number. This relationship can be seen by recognizing that \( \ln(e^x) = x \) and \( e^{\ln(x)} = x \).

• Understand that inverse functions reverse the effects of their counterparts.
• Function composition with inverses results in the identity function, meaning you get the original input back.
• The logarithmic-exponential relationship is pivotal to calculus and problem-solving involving natural logs and exponentials.

Definite Integral

The definite integral is a mathematical concept that calculates the area under a curve defined by a function. For a continuous function \( f(x) \) over a closed interval \([a, b]\), the definite integral \( \int_{a}^{b} f(x) \, dx \) measures the total area between the function's graph and the \( x \)-axis from \( x = a \) to \( x = b \).

In the context of the natural logarithm, the definition \( \ln(x) = \int_{1}^{x} \frac{dt}{t} \) describes the area under the curve \( \frac{1}{t} \) from \( t = 1 \) to \( t = x \). This concept ensures that each \( x \) value output from the logarithm has a precise mathematical meaning related to area.

• The lower limit of integration starts at 1 because \( \ln(1) = 0 \).
• Understand that the function inside the integral, \( \frac{1}{t} \), is a key part of calculating the natural log.
• The definite integral evaluates the cumulative effect or sum, vital for solving problems involving total accumulations.

Exponential Functions

Exponential functions are mathematical functions of the form \( e^x \), where \( e \) is the base of the natural logarithm, approximately equal to 2.718. These functions are characterized by their rapid rate of change, which is proportional to their current value.

The relationship between logarithms and exponentials makes them inverses. When you perform exponential and logarithmic operations, they cancel each other out, much like how addition and subtraction do. This is important in calculus, particularly when solving equations involving growth and decay.

• Exponential growth means self-reinforcing increases, such as population growth.
• The inverse property, where \( e^{\ln(x)} = x \), shows how they counteract one another.
• Exponential functions are not just theoretical; they model real-world phenomena.