/*! 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 33 use a calculator to evaluate the... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 33

use a calculator to evaluate the logarithm. Round to three decimal places. $$ \log _{2} 48 $$

Short Answer

Expert verified
The value of \( \log _{2} 48 \) rounded to three decimal places is 5.585.

Step by step solution

01

Understand the log expression

The expression \( \log _{2} 48 \) is in logarithmic form. It can be translated into an exponential form. In general, \( \log _{b} a = x \) is the same as \( b^x = a \).
02

Transform into exponential form

Start by setting the equation \( \log _{2} 48 = x \). This is equivalent to the exponential equation \( 2^x = 48 \). Now, the task is to find the value of \( x \) that makes this equation true.
03

Calculate logarithm

Use a scientific calculator to calculate \( \log _{2} 48 \). Simply input 'log48/log2' into the calculator to get the equivalent value. Remember to round the result to three decimal places as instructed by the exercise.

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.

Exponential Form
When we talk about exponential form, we're looking at equations expressed in the format \( b^x = a \), where \( b \) is the base, \( x \) is the exponent, and \( a \) is the result. In our example, when we say \( \log_2 48 \), we can convert this into exponential form as \( 2^x = 48 \). This tells us what power \( 2 \) must be raised to in order to result in \( 48 \). Understanding exponential form helps us clearly see the relationship between these numbers. It makes complex problems like finding logarithms more intuitive by showing them as power equations.
Logarithmic Form
Logarithmic form uses logs to express the same relationship shown in exponential form. If you have the expression \( b^x = a \) in exponential form, the equivalent logarithmic form would be \( \log_b a = x \). Essentially, logarithms are the inverse operations of exponentiation. In our case of \( \log_2 48 \), we are asking: "What exponent must 2 be raised to, to get 48?" This notation helps simplify complex relationships by converting them into straightforward mechanisms to solve for power terms. Reading a logarithm tells you how many times the base must be multiplied by itself to reach the target number.
Calculator Usage
To solve logarithms efficiently, calculators are invaluable tools. For our example using \( \log_2 48 \), you'll usually find your calculator lacks a button for base-2 logarithms directly. Here's a trick: use the change of base formula, which states \( \log_b a = \frac{\log_c a}{\log_c b} \). Most calculators have a built-in \( \log_{10} \) (common log) or natural log \( \ln \) function. To find \( \log_2 48 \) using your calculator, input \( \frac{\log 48}{\log 2} \). This will give you the accurate value, letting you skip paper-heavy logarithmic calculations and allowing you to quickly find answers with full precision.
Rounding Numbers
Rounding numbers is the process of trimming a number to make it more manageable, balancing accuracy with simplicity. In our exercise, precision is demanded to three decimal places. After calculating a logarithm or any other value, you might get a longer number like 5.585618鈥� Rounding this number to three decimal places means you'll only look at the figure three spaces after the decimal point. If the next digit is 5 or higher, you round up. So, 5.585618 becomes 5.586. This rule applies to any decimal rounding: focus on the fourth digit to decide whether the third rounds up or stays the same. It's essential for keeping work neat and results communicable.

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

Agriculture The yield \(V\) (in pounds per acre) for an orchard at age \(t\) (in years) is modeled by \(V=7955.6 e^{-0.0458 / t}\) (a) Use a graphing utility to graph the function. (b) Determine the horizontal asymptote of the context of the function. Interpret its meaning in the context of the problem. (c) Find the time necessary to obtain a yield of 7900 pounds per acre.

Solve for \(x\) or \(t\) $$ 300 e^{-0.2 t}=700 $$

\$ 3000\( is invested in an account at interest rate \)r,$ compounded continuously. Find the time required for the amount to (a) double and (b) triple. $$ r=0.12 $$

Learning Theory Students in a mathematics class were given an exam and then retested monthly with equivalent exams. The average scores \(S\) (on a 100 -point scale) for the class can be modeled by \(S=80-14 \ln (t+1),\) \(0 \leq t \leq 12,\) where \(t\) is the time in months. (a) What was the average score on the original exam? (b) What was the average score after 4 months? (c) After how many months was the average score \(46 ?\)

True or False? Determine whether the statement is true or false given that \(f(x)=\ln x\). If it is false, explain why or give an example that shows it is false.$$ \text { If } f(x)<0, \text { then } 0
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Pure 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.