/*! 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 834 Construct the graph of \(\mathrm... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 26: Problem 834

Construct the graph of \(\mathrm{y}=\log _{2} \mathrm{x}\).

Short Answer

Expert verified
To construct the graph of \(y = \log_2{x}\), first plot some key points: (1,0), (2,1), and (0.5,-1). Then, connect these points with a smooth curve that approaches the y-axis but never touches it, keeps rising as x increases, and crosses the x-axis at the point (1,0). The resulting curve is the graph of \(y = \log_2{x}\).

Step by step solution

01

Understanding the logarithm function

A logarithm function is denoted as \(\log_{b}{x}\), where b is the base of the logarithm and x is the value we are taking the logarithm of. It is the inverse function of an exponential function, which means that if y = \(\log_{b}{x}\), then x = \(b^{y}\). The logarithm function will satisfy the following conditions: 1. The domain is all positive real numbers (x > 0) 2. The range is all real numbers, i.e., (-鈭�, +鈭�) 3. If y = \(\log_{b}{x}\), then x = \(b^{y}\) 4. \(\log_{b}{1} = 0\), which means the graph always passes through the point (1,0). We will use these properties to plot the graph of y = \(\log_{2}{x}\).
02

Plot key points of the logarithm function

We will now plot some key points of the logarithm function using our knowledge of logarithms and the corresponding exponential functions: 1. When x = 1, y = \(\log_{2}{1} = 0\). This gives us the point (1,0). 2. When y = 0, x = \(2^{0} = 1\). This gives us the point (1,0) again. 3. When y = 1, x = \(2^{1} = 2\). This gives us the point (2,1). 4. When y = -1, x = \(2^{-1} = 0.5\). This gives us the point (0.5,-1). These key points will help us sketch the graph of the logarithm function.
03

Sketch the curve of the logarithm function

Now that we have some key points, we will connect these points with a smooth curve to get the graph of y = \(\log_{2}{x}\). The graph will resemble the following characteristics: 1. It will pass through the points (1,0), (0.5,-1), and (2,1). 2. The curve will approach the y-axis but never touch or cross it, as there is no value of y for x = 0 in the logarithm function. 3. The curve will continue to rise as x increases, approaching positive infinity. 4. The curve will cross the x-axis at the point (1,0), as \(\log_{b}{1} = 0\). After drawing the curve with the characteristics mentioned above, we will have the graph of y = \(\log_{2}{x}\).

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 \(\log _{3}\left(x^{2}-8 x\right)=2\).

Find \(5 \sqrt{2}\).

If \(\log \mathrm{N}=8.35721-10\), find \(\mathrm{N}\).

Find log \(_{3} 729 .\)

Graph the following functions: \((A) y=2^{x}\), (B) \(y=4^{\pi}\), (C) \(4^{-x}\), (D) \(\mathrm{y}=3 \cdot 2^{\mathrm{x}}\).
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

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.