/*! 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 43 Find the domain, \(x\) -intercep... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 43

Find the domain, \(x\) -intercept, and vertical asymptote of the logarithmic function and sketch its graph. $$y=\log \left(\frac{x}{7}\right)$$

Short Answer

Expert verified
The domain is \(x > 0\), the x-intercept is \(x = 7\), and the vertical asymptote is \(x = 0\).

Step by step solution

01

Determine the Domain

The domain of a logarithmic function includes only positive values, since we can't take a logarithm of a negative number or zero. Since x is divided by 7 in the logarithm of this function, the domain will be all x such that \(x / 7 > 0\), which simplifies to \(x > 0\). So, the domain is \(x > 0\).
02

Find the x-Intercept

The x-intercept of a function is the value of x when y=0. To find it, we set \(y = 0\) in our function and solve for x:\[0 = \log (\frac{x}{7}) \Rightarrow \frac{x}{7}=1 \Rightarrow x=7\]So, the x-intercept is \(x = 7\).
03

Determine the Vertical Asymptote

For a logarithmic function, a vertical asymptote exists at the value of x where the logarithm becomes undefined. Considering the domain, we can conclude that the function becomes undefined when \(x = 0\), so, the vertical asymptote is \(x = 0\).
04

Sketch the Graph

To sketch the graph of the logarithmic function:1. Plot the x-intercept at \(x = 7\).2. Draw the vertical asymptote at \(x = 0\).3. Sketch the curve of the function, making sure it approaches the vertical asymptote as \(x\) approaches 0 from the right and rises indefinitely as \(x\) increases.

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!

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

Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. $$\frac{1-\ln x}{x^{2}}=0$$

Use the acidity model given by \(\mathrm{pH}=-\log \left[\mathrm{H}^{+}\right],\) where acidity \((\mathrm{pH})\) is a measure of the hydrogen ion concentration \(\left[\mathrm{H}^{+}\right]\) (measured in moles of hydrogen per liter) of a solution. Compute \(\left[\mathrm{H}^{+}\right]\) for a solution in which \(\mathrm{pH}=5.8\).

Automobiles are designed with crumple zones that help protect their occupants in crashes. The crumple zones allow the occupants to move short distances when the automobiles come to abrupt stops. The greater the distance moved, the fewer g's the crash victims experience. (One \(g\) is equal to the acceleration due to gravity. For very short periods of time, humans have withstood as much as 40 g's.) In crash tests with vehicles moving at 90 kilometers per hour, analysts measured the numbers of g's experienced during deceleration by crash dummies that were permitted to move \(x\) meters during impact. The data are shown in the table. A model for the data is given by \(y=-3.00+11.88 \ln x+(36.94 / x),\) where \(y\) is the number of g's. $$ \begin{array}{|c|c|} \hline x & \text { g's } \\ \hline 0.2 & 158 \\ 0.4 & 80 \\ 0.6 & 53 \\ 0.8 & 40 \\ 1.0 & 32 \\ \hline \end{array} $$ (a) Complete the table using the model. $$ \begin{array}{|l|l|l|l|l|l|} \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 \\ \hline y & & & & & \\ \hline \end{array} $$ (b) Use a graphing utility to graph the data points and the model in the same viewing window. How do they compare? (c) Use the model to estimate the distance traveled during impact if the passenger deceleration must not exceed \(30 \mathrm{~g}\) 's. (d) Do you think it is practical to lower the number of g's experienced during impact to fewer than \(23 ?\) Explain your reasoning.

The number \(N\) of trees of a given species per acre is approximated by the model \(N=68\left(10^{-0.04 x}\right), 5 \leq x \leq 40,\) where \(x\) is the average diameter of the trees (in inches) 3 feet above the ground. Use the model to approximate the average diameter of the trees in a test plot when \(N=21\).

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\ln x+\ln (x+3)=1$$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

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.