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Chapter 2: Problem 116

If \(y=\log _{x} x\) and \(S=(3,27)\), the set onto which the set \(S\) is mapped is (A) \((0,3)\) (B) \((1,4)\) (C) \((1,3)\) (D) \((0,2)\)

Short Answer

Expert verified
The mapped set is always 1 for any x in (3,27), but no option directly fits this result as a range.

Step by step solution

01

Understanding the Function

The function given is \( y = \log_{x} x \). The logarithm of a number at the same base is 1, hence \( y = 1 \). This means for any \( x > 0 \), \( \log_{x} x = 1 \).
02

Analyzing the Given Set

We have the set \( S = (3,27) \). This means we are considering two values of \( x \), which are 3 and 27, both greater than zero.
03

Applying the Function to the Set

For any \( x \) within the set \( S = (3,27) \), substituting into the function gives \( y = \log_{x} x = 1 \). Thus, each element in set \( S \) maps to the value 1.
04

Determining the Mapped Set

Since the value \( y = 1 \) is constant regardless of the choice of \( x \) within \( S \), each element in the interval \( (3,27) \) maps to 1. Hence, the mapped set is \( \{1\} \).
05

Identify the Correct Answer

None of the given options (A, B, C, D) directly lists \( \{1\} \) since they represent ranges. Given our mapped set is a single value 1, the best representation from a range perspective must reflect only the possible outcome based on context, which is \((1,1)\), not directly listed.

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Key Concepts

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

Logarithmic Function
Logarithmic functions are a type of mathematical function used to solve problems relating to exponential growth and decay, among other things. A logarithmic function is expressed as \( y = \log_{b} a \), where \( b \) is the base, \( a \) is the argument, and \( y \) is the result or exponent. In the given problem, the specific function \( y = \log_{x} x \) simplifies to 1 because the log base of a number at itself is always 1.
\(
\)Understanding logarithmic functions can be broken down into:
  • Base: The base \( x \) in this context means that we are examining the relationship between a number and its exponent when expressed in terms of itself.
  • Basic Rule: Any number to the power of itself gives the power 1 (i.e., \( x^1 = x \)).
  • Simplification: For any real number \( x > 0 \), \( \log_{x} x = 1 \).
This concept is integral in mapping scenarios where outputs from a function are uniform across a range of inputs.
Mapping of Sets
Mapping of sets refers to applying a function to each element of a set and observing the results. It is often used to understand how a function changes or affects a set of inputs.
\(
\)In this specific exercise, we have a set \( S = (3,27) \) and are tasked with understanding what happens when the function \( y = \log_{x} x \) is applied to this set.
  • Initial Set: In our problem, the initial set \( S \) contains numbers greater than 0, specifically between 3 and 27.
  • Function Application: When applying \( \log_{x} x \) to each element of \( S \), every value maps to the result of 1 as discussed earlier.
  • Result of Mapping: Since each element of \( S \) results in the same value, it creates a simplified mapped set, \( \{1\} \).
This uniform result is due to the function's qualities and the nature of logarithms.
Logarithmic Equations
Logarithmic equations are equations involving logarithms, which can often involve solving for an unknown that is part of a logarithmic expression. This particular exercise allowed us to explore a simple logarithmic equation, \( y = \log_{x} x = 1 \).
\(
\)Exploration of logarithmic equations involves:
  • Equation Structure: Most equations of this nature compare a logarithm to a constant (e.g., \( \log_{b} a = c \)), where solving involves finding the value of \( a \) given \( b \) and \( c \).
  • Solving Approach: Using the property that \( \log_{x} x = 1 \), the equation is instantly solvable, as the exponent required to achieve the base is 1.
  • Recognizing Simplifications: In the context of this problem, it's insightful to recognize simple transformations and ensure correct mapping, as sometimes typical set expressions may not be evident.
With logarithms, a solid grasp of their basic properties and rules can help unravel more complex solutions and mappings.

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