/*! 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 5 Without using technology, find t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 5

Without using technology, find two consecutive whole numbers, \(a\) and \(b\) such that \(a<\log _{2} 28

Short Answer

Expert verified
4 and 5

Step by step solution

01

- Understand the problem

The goal is to find two consecutive whole numbers, say a and b, such that the logarithm of 28 to the base 2 lies between them: \(a < \log_{2} 28 < b\).
02

- Recall properties of logarithms

Recall that \(\log_{2} x\) represents the exponent to which 2 must be raised to get x. For instance, \(\log_{2} 8 = 3\) because \(2^3 = 8\) and \(\log_{2} 32 = 5\) because \(2^5 = 32\).
03

- Find bounds

Determine which powers of 2 are close to 28. Calculate powers around 28:i. \(2^4 = 16\) and ii. \(2^5 = 32\).
04

- Determine the inequality

From step 3, since \(2^4 = 16\) and \(2^5 = 32\), it follows that \(4 < \log_{2} 28 < 5\).
05

- Identify the consecutive integers

The consecutive integers \(a\) and \(b\) are 4 and 5, respectively. Therefore, \(a < \log_{2} 28 < b\) is true for \(a = 4\) and \(b = 5\).

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.

properties of logarithms
Logarithms are powerful tools in mathematics. They help us understand exponential relationships. The logarithm of a number to any base is the exponent to which the base must be raised to obtain that number. For example, for \(\text{log}_2 16\) we ask, '2 raised to what power equals 16?' The answer is 4 because \[2^4 = 16\]. This property of logarithms is crucial for solving problems involving exponential relationships.
logarithms and exponents
Logarithms and exponents are closely related. When you see \(\text{log}_b x\), you're looking for the power of \(\text{b}\) that gives you \text{x}. The connection is \[b^{\text{log}_bx} = x\]. Let's look at an example. Suppose we solve \(\text{log}_2 28\). We need to find the whole numbers \(\text{a}\) and \(\text{b}\) such that \[a < \text{log}_228 < b\]. We know \[2^4 = 16\] and \[2^5 = 32\], and since 28 is between 16 and 32, it follows that: \[4 < \text{log}_2 28 < 5\]. Logarithms translate the multiplication of numbers into the addition of their logarithms. This property makes them handy for simplifying calculations.
inequalities
Inequalities are mathematical expressions showing the relative sizes or values of two numbers or expressions. When dealing with logarithmic inequalities, we often compare the logarithm of a given number to different bases. For instance, in our original exercise, we needed to find two consecutive whole numbers \(\text{a}\) and \[\text{b}\] such that: \[a < \text{log}_2 28 < b\]. To do this, we calculated the bounds of the logarithm. We knew \[2^4 = 16\] and \[2^5 = 32\], placing \(\text{log}_2 28\) between 4 and 5. This helped us establish the inequality: \[\text{4} < \text{log}_2 28 < \text{5}\]. Understanding inequalities is crucial to accurately frame and solve such problems.

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

a) Only a vertical translation has been applied to the graph of \(y=\log _{3} x\) so that the graph of the transformed image passes through the point (9, - 4). Determine the equation of the transformed image. b) Only a horizontal stretch has been applied to the graph of \(y=\log _{2} x\) so that the graph of the transformed image passes through the point \((8,1) .\) Determine the equation of the transformed image.

The largest lake lying entirely within Canada is Great Bear Lake, in the Northwest Territories. On a summer day, divers find that the light intensity is reduced by \(4 \%\) for every meter below the water surface. To the nearest tenth of a meter, at what depth is the light intensity \(25 \%\) of the intensity at the surface?

Swedish researchers report that they have discovered the world's oldest living tree. The spruce tree's roots were radiocarbon dated and found to have \(31.5 \%\) of their carbon-14 (C-14) left. The half-life of C-14 is 5730 years. How old was the tree when it was discovered?

The compound interest formula is \(A=P(1+i)^{n},\) where \(A\) is the future amount, \(P\) is the present amount or principal, \(i\) is the interest rate per compounding period expressed as a decimal, and \(n\) is the number of compounding periods. All interest rates are annual percentage rates (APR). a) David inherits \(\$ 10\) 000 and invests in a guaranteed investment certificate (GIC) that earns \(6 \%,\) compounded semi-annually. How long will it take for the GIC to be worth \$11 000? b) Linda used a credit card to purchase a S1200 laptop computer. The rate of interest charged on the overdue balance is \(28 \%\) per year, compounded daily. How many days is Linda's payment overdue if the amount shown on her credit card statement is \(\$ 1241.18 ?\) c) How long will it take for money invested at \(5.5 \%,\) compounded semi- annually, to triple in value?

If \(\log _{3} 81=x-y\) and \(\log _{2} 32=x+y\) determine the values of \(x\) and \(y\)
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Logic and Functions

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.