Problem 383 - 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. Use the identity \(\ln (x)=\int_{1}^{x} \frac{d t}{t}\) to derive the identity \(\ln \left(\frac{1}{x}\right)=-\ln x.\)

Step-by-Step Solution

Answer

\(\ln\left(\frac{1}{x}\right) = -\ln x\) by substitution and simplifying integrals.

Step 1: Express a Definite Integral

Starting from the given identity, express \(\ln(1/x)\) as a definite integral: \[\ln\left(\frac{1}{x}\right) = \int_{1}^{1/x} \frac{dt}{t}.\]

Step 2: Use Substitution to Transform the Integral

Substitute \(t = \frac{1}{u}\), which implies \(dt = -\frac{1}{u^2} du\). The limits of integration also change: when \(t = 1\), \(u = 1\), and when \(t = \frac{1}{x}\), \(u = x\). Thus, the integral transforms to: \[\int_{1}^{1/x} \frac{dt}{t} = \int_{1}^{x} -\frac{1}{u} du.\]

Step 3: Simplify the Integral

Rewrite the integral with the negative sign pulled out: \[-\int_{1}^{x} \frac{du}{u}.\] This is equivalent to \[-\ln(x)\] by the definition of the natural logarithm.

Step 4: Conclude the Equality

Thus, by rewriting and simplifying the definite integral, we've shown that \[\ln\left(\frac{1}{x}\right) = -\ln x.\] This is the identity we needed to establish.

Key Concepts

Definite Integrals

The concept of definite integrals plays a fundamental role in understanding how natural logarithms can be expressed and manipulated through calculus. A definite integral is a specific kind of integral that calculates the accumulated area under a curve \(y = f(t)\) from a lower limit to an upper limit. In the context of natural logarithms, the integral \(( x) =\int_{1}^{x} \frac{dt}{t}\) is used. This equation represents the area under the curve of \(\frac{1}{t}\) from \(t=1\) to \(t=x\).

• The limits specify where the integration begins and ends.
• The value of a definite integral depends on the limits and the function being integrated over these limits.

Here, the definite integral from 1 to \(x\) helps define how logarithmic functions are accumulated through an area under the curve, linking the integral calculation to the value of \(\ln(x)\). This foundational concept connects calculus with logarithmic identities.

Integration by Substitution

Integration by substitution is a technique used to solve integrals that are not straightforward. It's similar to reversing the chain rule. In this exercise, we used substitution to transform the integral for \(\ln\left(\frac{1}{x}\right)\). By substituting \(t = \frac{1}{u}\), we effectively "re-map" the integral into simpler forms. Here's how it works:

• We choose a substitution that simplifies the integral.
• In this case, substituting \(t = \frac{1}{u}\) transforms \(dt\) into \(-\frac{1}{u^2}du\).
• The limits of integration also change according to the substitution.

After substitution, the integral \(\int_{1}^{1/x} \frac{dt}{t}\) becomes \[\int_{1}^{x} -\frac{1}{u} du.\]\ . This new integral is often simpler or easier to evaluate.
This technique is invaluable in solving complex integrals that come across in calculus, providing a structured method to address new and challenging forms.

Logarithmic Identities

Logarithmic identities are rules that simplify the manipulation and understanding of logarithms. In our case, we derived one such identity, \(\ln\left(\frac{1}{x}\right) = -\ln(x)\), using definite integrals and substitution. Understanding this identity helps in many areas of math, especially in equations involving exponential growth or decay.

• The identity tells us that the logarithm of the reciprocal of a number is equal to the negative of the logarithm of the number.
• This property becomes useful when dealing with reciprocals in computations or simplifying expressions.

Through this derived identity, we deepen our understanding of how logarithms work and why they behave the way they do, making our interactions with exponential functions much smoother.

Properties of Logarithms

Logarithms have several properties that make them versatile tools for calculations and problem-solving. These properties allow us to simplify complex expressions and solve equations involving exponential terms. An important property related to our exercise is the relationship between logarithms and division, which is elegantly summarized by \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\).

• This indicates how division inside the logarithm becomes subtraction outside.
• The negative sign in \(\ln\left(\frac{1}{x}\right)\) reflects this division/subtraction relationship.

Overall, understanding these properties allows for manipulation and simplification of logarithmic expressions, facilitating solutions to algebraic problems. Recognizing these patterns and rules regarding logarithms is critical for mastering higher-level mathematics.