/*! 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 27 Find the exact value of each log... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 27

Find the exact value of each logarithm without using a calculator. $$ \log _{2} 1 $$

Short Answer

Expert verified
The value of \(\log_{2} 1\) is 0.

Step by step solution

01

- Understand the Problem

We need to find the exact value of the logarithm \(\log_{2}1\). This means we are looking for the exponent \(x\) such that \({2}^{x} = 1\).
02

- Recall the Property of Logarithms

Recall that any number (except zero) raised to the power of 0 is 1. That is, \[a^0 = 1\].
03

- Apply the Property to the Given Base

Apply this property to the base 2: \[2^0 = 1\]. Therefore, the exponent \(x\) must be 0.
04

- State the Answer

Since \(\log_{2}1\) is the exponent that 2 must be raised to in order to get 1, and we established that \(2^0 = 1\), the value of \(\log_{2}1\) is 0.

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.

Logarithm Properties
Logarithms have unique properties that make them vital in algebra and higher mathematics. One essential property is that any non-zero number raised to the power of zero is always 1. This can be expressed as: \[ a^0 = 1 \].
For example: \ \( 5^0 = 1 \), \( 10^0 = 1 \).
Another useful property is the logarithm of 1 with any base is always 0. Thus: \[ \log_a 1 = 0 \].
This is because any number raised to the power of 0 will always result in 1, confirming similar behavior across different bases such as:\br>\( \log_{5} 1 = 0 \), \[ \log_{10}1 = 0 \].
Remembering these properties can simplify many logarithmic calculations and save time.
Logarithm Base
The base of a logarithm is the number that is raised to a power. For example, in \( \log_{2}1 \), the base is 2. The base determines the system in which the logarithm operates.

Common bases include 10 (common logarithm, commonly denoted as \( \log \)), and Euler's number e (natural logarithm, denoted as \( \ln \)).

The base helps to reconstruct the equation in exponential form. For \( \log_{2}1 \), we ask: '2 raised to what power equals 1?' In mathematical terms: \ \(2^x = 1 \).' Knowing this, we can now understand logarithms as inverse operations to exponents. The correct exponent here is 0.
Exponents
Understanding exponents is crucial when working with logarithms. Exponents indicate how many times a base is multiplied by itself. For example, \( 2^3 \) means \( 2 \) is multiplied 3 times: so \( 2 * 2 * 2 \).
Common exponent rules include:
  • Product of powers: \[ a^m * a^n = a^{m+n} \]
  • Power of a power: \[ (a^m)^n = a^{m*n} \]
  • Zero exponent rule: \[ a^0 = 1 \]
  • Negative exponent rule: \[ a^{-n} = \frac{1}{a^n} \]

Understanding these rules, helps solve logarithmic problems more effectively. For instance, finding \( \log_{2}1 \) requires applying \( a^0 = 1 \) rule. Recognize that any base raised to 0 is 1, thereby clarifying that point in our logarithmic context.

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

Express y as a function of \(x .\) The constant \(C\) is a positive number. \(\ln y=-2 x+\ln C\)

Suppose that \(G(x)=\log _{3}(2 x+1)-2\) (a) What is the domain of \(G ?\) (b) What is \(G(40) ?\) What point is on the graph of \(G ?\) (c) If \(G(x)=3,\) what is \(x ?\) What point is on the graph of \(G ?\) (d) What is the zero of \(G ?\)

Solve each equation. $$ \log _{4} 64=x $$

Express y as a function of \(x .\) The constant \(C\) is a positive number. \(\ln y=3 x+\ln C\)

Use the Change-of-Base Formula and a calculator to evaluate each logarithm. Round your answer to three decimal places. \(\log _{\pi} e\)
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Mechanics 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.