Problem 386 - Calculus Volume 1

Question

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

We deduce that \( E(a+b) = E(a)E(b) \) using the property of logarithms.

Step 1: Understand the given identity

We are provided with the property of natural logarithms: \( \ln(xy) = \ln(x) + \ln(y) \). This means if you have two numbers \( x \) and \( y \), the logarithm of their product is equal to the sum of their logarithms. We will use this identity as a base to find out more about \( E \) function.

Step 2: Define the inverse relationship

Since \( E \) is the inverse of the natural logarithm function \( \ln \), we have \( E(\ln(x)) = x \) for any positive real number \( x \). This property of inverses means that \( E \) applied to \( \ln \) of a number gives us the original number back.

Step 3: Explore substitution using \( E \) function

Let's consider \( a = \ln(x) \) and \( b = \ln(y) \). Then we have \( E(a) = x \) and \( E(b) = y \). According to the logarithmic identity, \( a + b = \ln(x) + \ln(y) = \ln(xy) \).

Step 4: Apply the substitution result to \( E \) function

Now using the property of \( E \), we find \( E(a + b) = E(\ln(xy)) \). Since \( E \) is the inverse of \( \ln \), \( E(\ln(xy)) = xy \). Hence, \( E(a + b) = E(a) \cdot E(b) = xy \).

Step 5: Conclude the deduction

We have determined from the steps above that \( E(a + b) = E(a) \cdot E(b) \) for any real numbers \( a \) and \( b \). This property of \( E \) mirrors that of the exponential function, confirming the multiplicative nature.

Key Concepts

Logarithmic Identity

The logarithmic identity is an essential property of logarithms, which states that the logarithm of a product is equal to the sum of the logarithms of the individual numbers. Mathematically, it can be represented as: \( \ln(xy) = \ln(x) + \ln(y) \). This identity shows how logarithms can simplify multiplication into addition, making calculations easier.

When solving problems involving inverse functions, this logarithmic identity becomes crucial. For instance, if we have an inverse function \( E \) and we know \( E \) is the inverse of the natural logarithm, this identity helps establish relationships between values within the function.

Key uses of this identity:

• It allows the decomposition of complex algebraic expressions into simpler ones.
• Provides foundations for solving equations involving exponential growth or decay.
• Essential for deducing properties of functions like \( E \), helping understand inverse relationships.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a fundamental logarithmic function with the base \( e \) (approximately 2.71828). It is predominantly used in calculus due to its unique properties and its natural occurrence in mathematical modeling.

The natural logarithm has several distinct features:

• It converts multiplication into addition, a property derived from the logarithmic identity \( \ln(xy) = \ln(x) + \ln(y) \).
• The derivative of the natural logarithm function is \( \frac{1}{x} \), which appears frequently in solving differential equations.
• Any function that can be written as \( e^{y} \) has \( y = \ln(x) \) as its inverse function, showcasing its role in reversing exponential functions.

In the context of inverse functions, the natural logarithm helps define the function \( E \). If \( E \) is the inverse of \( \ln \), it would return the original value, i.e., \( E(\ln(x)) = x \). This principle is crucial in connecting logarithmic and exponential functions.

Exponential Function

An exponential function is a mathematical function of the form \( f(x) = a^{x} \), where \( a \) is a constant. The most common base for these functions in calculus is \( e \), leading to \( f(x) = e^{x} \). This expression not only represents rapid growth but is also widely used due to its unique and defining relationship with natural logarithms.

Exponential functions have key characteristics:

• The derivative of \( e^{x} \) is itself, \( e^{x} \), which simplifies many calculus operations.
• They are the inverse of natural logarithms, meaning if \( f(x) = e^{x} \), then its inverse \( f^{-1}(x) = \ln(x) \).
• The property \( e^{a + b} = e^{a} \, e^{b} \) aligns perfectly with the logarithmic identity.

This multiplicative property noted in the original exercise indicates why the function \( E \) behaves like \( e^{x} \). By demonstrating that \( E(a+b) = E(a) \cdot E(b) \), we recognize \( E \) as having characteristics similar to an exponential function. This strong relationship helps solve equations and model real-world phenomena efficiently.