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Chapter 4: Problem 27

In Exercises 21鈥�42, evaluate each expression without using a calculator. $$ \log _{2} \frac{1}{8} $$

Short Answer

Expert verified
The value of \( \log _{2} \frac{1}{8} \) is -3

Step by step solution

01

Identify the Problem

We are asked to evaluate \( \log _{2} \frac{1}{8} \). The base of this logarithm is 2 and the argument (value inside the logarithm) is 1/8.
02

Rewrite the fraction as a power

Recall that any fraction like 1/n is equivalent to n raised to the power of -1. So, we can rewrite 1/8 as \(2^{-3}\). Thus, the problem \( \log _{2} \frac{1}{8} \) became \( \log _{2}{(2^{-3})} \).
03

Apply Power Rule of Logarithms

The power rule of logarithms states that \( \log _{b}{(a^{n})} = n \cdot log_{b}{a} \). Applying it to our expression, \( \log _{2}{(2^{-3})} \) became \(-3 \cdot log_{2}{2}\).
04

Evaluate the Logarithm

Remember that any logarithm base b of b is equals to 1. Therefore, the expression \(-3 \cdot log_{2}{2}\) becomes \(-3 \cdot 1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithmic Properties
Understanding logarithmic properties can simplify various mathematical expressions involving logarithms. These properties allow us to manipulate logarithms in ways that can make solving problems more straightforward. Here, we'll cover the key properties used in our exercise.

  • **Product Rule**: \( \log_{b} (xy) = \log_{b} x + \log_{b} y \)
  • **Quotient Rule**: \( \log_{b} \left(\frac{x}{y}\right) = \log_{b} x - \log_{b} y \)
  • **Power Rule**: \( \log_{b} (x^n) = n \cdot \log_{b} x \)

In the given exercise, we effectively use the power rule to simplify the expression by turning the exponentiation inside the argument of the logarithm into a multiplication outside of it. Applying these rules can turn a seemingly complex logarithmic problem into something much more manageable. Let's explore each one more deeply.
Exponents
Exponents represent the number of times a number (the base) is multiplied by itself. They are a crucial component in mathematics and provide a concise way to express repeated multiplication.

**Definition of Exponents**:
  • The expression \(a^n\) signifies that the base \(a\) is multiplied by itself \(n\) times.
  • Inversely, a negative exponent indicates division, turning the base into its reciprocal.

In our exercise, understanding exponents is critical because \(\frac{1}{8}\) has been rewritten as \(2^{-3}\). This is achieved because as \(2^3 = 8\), the reciprocal is expressed by reversing the sign of the exponent, giving us \(2^{-3}\), highlighting both multiplication by itself and the concept of inversion which are fundamental to exponents.
Base of Logarithm
The base of a logarithm is a foundational concept in understanding how to compute logarithmic expressions. The base determines what number the power is based upon.

**Understanding the Base**:
  • The base is the small number found here in the subscript: \log_{b} x\. In \log_{2} \, 2 is the base.
  • Logarithms are essentially asking the question: "To what power must the base be raised, to produce a given number?"
  • For example, in \(\log_{2} 8\), this is asking which power of 2 gives you 8? The answer is 3, since \(2^3 = 8\).

In the original exercise, knowing that the base is 2 helps us evaluate \(\log_{2}(2^{-3}) \) directly to \ -3 \, because we recognize that the argument is a power of the base. Hence, making evaluations like these much simpler when the base and the argument align accordingly.

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Most popular questions from this chapter

In Example I on page \(520,\) we used two data points and an exponential function to model the population of the United States from 1970 through 2010 . The data are shown again in the table. Use all five data points to solve Exercises \(66-70\). $$ \begin{array}{cc} {x, \text { Number of Years }} & {y, \text { U.S. Population }} \\ {\text { after } 1969} & {\text { (millions) }} \\ {1(1970)} & {203.3} \\ {11(1980)} & {226.5} \\ {21(1990)} & {248.7} \\ {31(2000)} & {281.4} \\ {41(2010)} & {308.7} \end{array} $$ Use your graphing utility's logarithmic regression option to obtain a model of the form \(y=a+b \ln x\) that fits the data. How well does the correlation coefficient, \(r,\) indicate that the model fits the data?

In Exercises \(128-131\), graph \(f\) and \(g\) in the same viewing rectangle. Then describe the relationship of the graph of \(g\) to the graph of \(f\) $$ f(x)=\log x, g(x)=-\log x $$

In many states, a \(17 \%\) risk of a car accident with a blood alcohol concentration of 0.08 is the lowest level for charging a motorist with driving under the influence. Do you agree with the \(17 \%\) risk as a cutoff percentage, or do you feel that the percentage should be lower or higher? Explain your answer. What blood alcohol concentration corresponds to what you believe is an appropriate percentage?

The exponential growth models describe the population of the indicated country, \(A,\) in millions, t years after 2006 . $$ \begin{array}{ll} {\text { Canada }} & {A=33.1 e^{0.009 t}} \\ {\text { Uganda }} & {A=28.2 e^{0.034 t}} \end{array} $$ Use this information to determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. In \(2006,\) Canada's population exceeded Uganda's by 4.9 million.

Exercises 150鈥�152 will help you prepare for the material covered in the next section. In each exercise, evaluate the indicated logarithmic expressions without using a calculator. a. Evaluate: \(\log _{2} 16\) b. Evaluate: \(\log _{2} 32-\log _{2} 2\) c. What can you conclude about $$ \log _{2} 16, \text { or } \log _{2}\left(\frac{32}{2}\right) ? $$
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