/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 85 Graph \(\ln x, \ln 2 x, \ln 4 x,... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 85

Graph \(\ln x, \ln 2 x, \ln 4 x, \ln 8 x,\) and \(\ln 16 x\) (as many as you can) together for \(0 < x \leq 10 .\) What is going on? Explain.

Short Answer

Expert verified
Each graph is a vertical translation of \( \ln x \) by a constant \( \ln n \), representing horizontal stretching.

Step by step solution

01

Analyze the Functions

The functions to graph are \( \ln x \), \( \ln 2x \), \( \ln 4x \), \( \ln 8x \), and \( \ln 16x \). These can be rewritten using logarithmic properties as \( \ln x + \ln 2 \), \( \ln x + \ln 4 \), \( \ln x + \ln 8 \), and \( \ln x + \ln 16 \).
02

Understanding Vertical Shifts

Recognize that each function \( \ln nx \) is simply \( \ln x \) vertically shifted by \( \ln n \). The shifts are constant and do not affect the shape of the graph, only its vertical position. More specifically: \( \ln 2x \) shifts \( \ln 2 \) units up, \( \ln 4x \) shifts \( \ln 4 \) units up, and so on.
03

Sketch the Graphs

Plot the graph of \( \ln x \). Then, draw each subsequent function \( \ln 2x \), \( \ln 4x \), \( \ln 8x \), and \( \ln 16x \) by shifting the graph of \( \ln x \) vertically up by \( \ln 2, \ln 4, \ln 8, \) and \( \ln 16 \), respectively. Ensure that all graphs are plotted over the interval \( 0 < x \leq 10 \).
04

Interpretation

The graphs are vertically shifted above each other. Each function's graph is a vertical translation of the previous one, with the same shape and slope since the derivative \( \frac{1}{x} \) of \( \ln x \) remains unchanged.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Vertical Shifts in Logarithmic Functions
Vertical shifts occur when the entire graph of a function is moved up or down on a coordinate plane. When dealing with logarithmic functions, vertical shifts don't alter the function's base shape or slope. For instance, in the functions given in the exercise,
  • \( \ln x \) is the base graph.
  • When you look at \( \ln 2x \), \( \ln 4x \), etc., these expressions can be rewritten using the property of logarithms: \( \ln nx = \ln x + \ln n \), showing that each graph is simply that of \( \ln x \) shifted vertically by \( \ln n \).
  • Thus, \( \ln 2x \) would be \( \ln x \) shifted vertically upward by \( \ln 2 \), \( \ln 4x \) by \( \ln 4 \), and so on.
These shifts are constant; they do not stretch or compress the graph, meaning they maintain the same vertical distance throughout the entire range of x-values. This causes a series of parallel vertical translations.
Graphing Logarithmic Functions
Graphing logarithmic functions involves understanding the unique shape and properties of these graphs. The graph of the natural logarithm function \( \ln x \) is the blueprint for other logarithmic functions like \( \ln 2x \), \( \ln 4x \), etc.
  • The graph has a vertical asymptote along the y-axis, indicating it approaches but never touches this line.
  • It's continuously increasing, with a gentle upward curve as x values increase.
  • The slope becomes less steep as x grows larger, but always remains positive, showing that \( \ln x \) increases without bound.
When graphing functions like \( \ln 2x \) or \( \ln 4x \), simply start from the graph of \( \ln x \) and apply the vertical shift. Use this understanding to easily transition from one graph to the next.
Key Properties of Logarithms
Logarithms have several inherent properties that make manipulating and transforming them less daunting. Here are some important ones relevant to the exercise:
  • The product rule: \( \ln (ab) = \ln a + \ln b \), which is used in the exercise to turn \( \ln(nx) \) into \( \ln x + \ln n \). This shows how products inside the logarithm can be split into a sum of logs.
  • The change of base property, while not directly used here, is valuable to know: \( \log_b a = \frac{\ln a}{\ln b} \). It allows converting logs from one base to another using natural logarithms.
  • The property of the logarithmic identity: \( \ln 1 = 0 \), which reflects that any logarithm of 1, no matter the base, equals zero.
Understanding these properties helps in simplifying expressions and solving equations involving logarithms, steering the way to effectively graph and interpret logarithmic functions. Each property contributes to a fuller comprehension of how to manage transformations such as shifts in a natural logarithm graph.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Evaluate the integrals in Exercises \(51-60 .\) $$ \int_{0}^{\ln 2} 4 e^{-\theta} \sinh \theta d \theta $$

Evaluate the integrals in Exercises \(41-50\) . $$ \int \tanh \frac{x}{7} d x $$

Find the limits in Exercises \(113-116\) $$ \lim _{x \rightarrow 0} \frac{2 \tan ^{-1} 3 x^{2}}{7 x^{2}} $$

Find the limits in Exercises \(113-116\) $$ \lim _{x \rightarrow 1^{+}} \frac{\sqrt{x^{2}-1}}{\sec ^{-1} x} $$

Evaluate the integrals in Exercises \(51-60 .\) $$ \int_{\ln 2}^{\ln 4} \operatorname{coth} x d x $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.