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Chapter 8: Problem 23

Evaluate each logarithm. $$ \log _{\frac{1}{2}} \frac{1}{2} $$

Short Answer

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

Step by step solution

01

Understand the logarithm

First, it's essential to understand what \(\log _{\frac{1}{2}} \frac{1}{2}\) is asking for. This logarithm asks the question 'To what power must we raise \(\frac{1}{2}\) to obtain \(\frac{1}{2}\)?'.
02

Apply the Identity Property of Logarithms

The identity property of logarithms states that the \(\log_b (b) = 1\). In this particular case \(b=\frac{1}{2}\). So applying identity property, \(\log _{\frac{1}{2}} \frac{1}{2} = 1\).
03

Final step

So, \(\log _{\frac{1}{2}} \frac{1}{2}\) equals 1, which can be interpreted as '\(\frac{1}{2}\) raised to the power of 1 equals \(\frac{1}{2}\)'.

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Key Concepts

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

Identity Property of Logarithms
The identity property of logarithms is a fundamental concept that simplifies working with logarithmic expressions. Essentially, this property tells us that if the base of the logarithm and the argument of the logarithm are the same, the result will always be 1.
  • Mathematically, it is expressed as \( \log_b(b) = 1 \).
  • This property answers the question of what exponent is needed to raise the base \(b\) to itself.
  • In the example \( \log_{\frac{1}{2}} \frac{1}{2} \), applying this property gives us 1 because \( \frac{1}{2}^1 = \frac{1}{2} \).
Understanding this property allows us to quickly evaluate logarithmic expressions when the base and the argument are identical.
Exponentiation
Exponentiation is a mathematical operation that involves raising a number, called the base, to the power of an exponent. It expresses repeated multiplication of the base.
  • If you have \( b^x \), \(b\) is the base, and \(x\) is the exponent.
  • For example, \( 2^3 = 2 \times 2 \times 2 = 8 \).
  • Exponentiation is integral to understanding logarithms, as a logarithm is the inverse operation of exponentiation.
In the exercise, when you see \( \frac{1}{2}^1 = \frac{1}{2} \), it is showing that raising the base \(\frac{1}{2}\) to the power of 1 results in the original number, which connects back to the identity property of logarithms.
Base of a Logarithm
The base of a logarithm is a critical component in understanding how logarithms work.
  • The base is the number that is raised to the power of the result of the logarithm.
  • In the logarithmic form \( \log_b(y) \), \(b\) is the base.
  • The base tells us the number we start with before applying the exponent to achieve the argument \(y\).
Having a clear understanding of what a base is in a logarithm helps when solving problems, especially when utilizing properties like the identity property. For example, in \( \log_{\frac{1}{2}} \frac{1}{2} \), the base is clearly \(\frac{1}{2}\), and it sets the context for identifying that the exponent must be 1 to satisfy the equation.

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Most popular questions from this chapter

Write an equation. Then solve the equation without graphing. Multiple choice As a town gets smaller, the population of its high school decreases by 12\(\%\) each year. The student body has 125 students now. In how many years will it have about 75 students? $$\begin{array}{llll}{\text { (A) } 4 \text { years }} & {\text { (B) } 7 \text { years }} & {\text { (C) } 10 \text { years }} & {\text { (D) } 11 \text { years }}\end{array}$$

Solve each equation. Check your answers. $$ 3 \log x=1.5 $$

Write each equation in logarithmic form. \(49=7^{2}\)

\(\log _{5} 10 \approx 1.4307\) and \(\log _{5} 20 \approx 1.8614 .\) Find the value of \(\log _{5}\left(\frac{1}{2}\right)\) without using a calculator. Explain how you found the value.

The equation \(y=281(1.0124)^{x}\) models the U.S. population \(y,\) in millions of people, \(x\) years after the year 2000 . Graph the function on your graphing calculator. Estimate when the U.S. population will reach 350 million.
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