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Chapter 6: Problem 26

WRITING Explain why the expressions \(\log _2(-1)\) and \(\log _1 1\) are not defined.

Short Answer

Expert verified
Both \(\log _2(-1)\) and \(\log _1 1\) are not defined because they violate one or the other basic property of the logarithmic function. \(\log _2(-1)\) isn't valid since it has a negative argument, and \(\log _1 1\) isn't valid because having 1 as the base contradicts the rule that the base should not be 1.

Step by step solution

01

Understanding base and arguments in logarithmic function

In a logarithmic function \(y = \log_a{b}\), 'a' is the base and 'b' is the argument. Logarithms are only defined for positive arguments and the base must be greater than 0 but not equal to 1.
02

Analyze the expression \(\log _2(-1)\)

In the expression \(\log _2(-1)\), '2' is the base which is positive. But the argument, \(-1\), is negative, which violates the rule that logarithmic functions are only defined for positive arguments. Therefore, \(\log _2(-1)\) is not defined.
03

Analyze the expression \(\log _1 1\)

In the expression \(\log _1 1\), though the argument '1' is positive, the base is also '1', which contradicts the rule that the base for logarithmic functions must be greater than 0 but not equal to 1. Therefore, \(\log _1 1\) is not defined.

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Key Concepts

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

Logarithms
Logarithms are often introduced as a way to solve for exponents. They are the inverse operations of exponentiation. Simply put, if you have an equation like \( a^y = b \), the logarithm helps you solve for \( y \). In this case, it would be expressed as \( y = \log_a{b} \). The goal of logarithms is to find out what exponent we need to raise the base \( a \) to, in order to get the number \( b \).
  • Logarithms represent exponents.
  • The expression \( \log_a b \) is read as "log base \( a \) of \( b \)".
  • Logarithms are only defined for positive numbers.
So, it's crucial that the argument (the \( b \) in \( \log_a b \)) is positive. Negative or zero arguments result in undefined expressions, as we can't have a power that yields a negative or zero value for real numbers.
Understanding these principles helps in realizing why expressions like \( \log_2(-1) \) are not defined.
Base of Logarithm
The base of a logarithm is the number that is raised to a power to achieve the argument. For a logarithm \( \log_a b \), \( a \) is the base. The choice of base affects the value of the logarithm greatly.
  • The base \( a \) should be greater than zero.
  • The base \( a \) should not be equal to 1.
Why can't the base equal 1? Well, think about \( 1^x \). No matter the value of \( x \), \( 1^x \) will always equal 1. Thus, a logarithm with base 1 would be meaningless, as it can't distinguish between different values of \( b \). This is why expressions like \( \log_1 1 \) aren't defined.
In most practical uses, we often see bases of 10 and \( e \) (Euler's number, approximately 2.718) since these bases are very useful in science and engineering.
Arguments of Logarithm
The argument of a logarithm is the number you are taking the logarithm of. In the expression \( \log_a{b} \), \( b \) is your argument. The argument must meet certain criteria for the logarithm to be defined.
  • The argument \( b \) must be greater than zero.
  • Arguments of zero or negative values are not possible.
Why is this the case? Consider this: if \( a^x = b \), and \( b \) was zero or negative, there would be no real number \( x \) such that \( a^x \) results in \( b \). This is because a positive base raised to any real number always results in a positive outcome.
This understanding highlights why expressions like \( \log_2(-1) \) are not defined because \(-1\) is a negative, hence invalid argument for logarithms. Grasping the importance of both base and expectation can clarify much of logarithmic functionality and restrictions.

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Most popular questions from this chapter

The apparent magnitude of a star is a measure of the brightness of the star as it appears to observers on Earth. The apparent magnitude \(M\) of the dimmest star that can be seen with a telescope is \(M=5 \log D+2\), where \(D\) is the diameter (in millimeters) of the telescope's objective lens. What is the diameter of the objective lens of a telescope that can reveal stars with a magnitude of \(12 ?\)

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