Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions

Problem 7.1.68

Chapter 7, Problem 7.1.68

Logarithm properties Use the integral definition of the natural logarithm to prove that ln(x/y) = ln x - ln y.

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Recall the integral definition of the natural logarithm: for any positive number $ a $, $ \ln a = \int_1^a \frac{1}{t} \, dt $.

Express $ \ln \left( \frac{x}{y} \right) $ using the integral definition: $ \ln \left( \frac{x}{y} \right) = \int_1^{\frac{x}{y}} \frac{1}{t} \, dt $.

Use a substitution to rewrite the integral with limits from 1 to $ x $ and 1 to $ y $. Let $ t = \frac{u}{y} $, so that when $ t = 1 $, $ u = y $, and when $ t = \frac{x}{y} $, $ u = x $.

Rewrite the integral in terms of $ u $: $ \int_1^{\frac{x}{y}} \frac{1}{t} \, dt = \int_y^x \frac{1}{\frac{u}{y}} \cdot \frac{1}{y} \, du = \int_y^x \frac{y}{u} \cdot \frac{1}{y} \, du = \int_y^x \frac{1}{u} \, du $.

Split the integral $ \int_y^x \frac{1}{u} \, du $ into $ \int_1^x \frac{1}{u} \, du - \int_1^y \frac{1}{u} \, du $, which equals $ \ln x - \ln y $, thus proving $ \ln \left( \frac{x}{y} \right) = \ln x - \ln y $.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Integral Definition of the Natural Logarithm

The natural logarithm ln(x) can be defined as the integral from 1 to x of 1/t dt. This definition provides a fundamental way to understand ln(x) using calculus, linking logarithms to areas under curves.

Properties of Definite Integrals

Definite integrals have properties such as additivity over adjacent intervals and the ability to split or combine integrals. These properties allow manipulation of integrals to express ln(x/y) in terms of ln(x) and ln(y).

Logarithm Laws and Their Proofs

Logarithm laws, like ln(x/y) = ln(x) - ln(y), describe relationships between logarithms of products and quotients. Using the integral definition and integral properties, these laws can be rigorously proven rather than assumed.

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