/*! 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 12 Evaluate the indicated expressio... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 12

Evaluate the indicated expression. Do not use a calculator for these exercises. $$ \log _{2} \frac{1}{256} $$

Short Answer

Expert verified
The short version of the answer is: When evaluating \(\log_{2}\frac{1}{256}\), we can rewrite the fraction as a power of 2: \(\frac{1}{256} = 2^{-8}\). Then, applying the properties of logarithms, we get \(\log_{2}(2^{-8}) = -8\), so the final answer is \(-8\).

Step by step solution

01

Express the fraction as a power of 2

First, we need to express the fraction \(\frac{1}{256}\) as a power of 2. Since \(256 = 2^8\), we can rewrite the fraction as: \[ \frac{1}{256} = \frac{1}{2^8} = 2^{-8} \]
02

Apply the properties of logarithms

Now that we have found the fractional value as a power of 2, we can plug it into our logarithmic expression \(\log_{2}\frac{1}{256}\) and use the properties of logarithms to solve it. Replace the fraction with its equivalent form in powers of 2: \[ \log_{2}(2^{-8}) \]
03

Solve the logarithmic expression

Since the base and the argument inside the logarithm are the same, the logarithm "cancels out" the exponential function, leaving us with just the exponent: \[ \log_{2}(2^{-8}) = -8 \] Therefore, the value of the expression \(\log_{2}\frac{1}{256}\) is \(-8\).

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

Show that $$ (\cosh x+\sinh x)^{t}=\cosh (t x)+\sinh (t x) $$ for all real numbers \(x\) and \(t\).

Find all numbers \(r\) such that \(\ln \left(2 r^{2}-3\right)=-1\).

Suppose \(b\) is a small positive number. Estimate the slope of the line containing the points \(\left(e^{3}, 5+b\right)\) and \(\left(e^{3+b}, 5\right)\).

What is wrong with the following apparent paradox: You have two parents, four grandparents, eight greatgrandparents, and so in. Going back \(n\) generations, you should have \(2^{n}\) ancestors. Assuming three generations per century, if we go back 2000 years (which equals 20 centuries and thus 60 generations), then you should have \(2^{60}\) ancestors from 2000 years ago. However, \(2^{60}=\left(2^{10}\right)^{6} \approx\left(10^{3}\right)^{6}=10^{18},\) which equals a billion billion, which is far more than the total number of people who have ever lived.

Find a number \(x\) such that \(\ln x=-2\).
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

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.