What Are The Domain And Range Of $f(x) = \log X - 5$?A. Domain: $x \ \textgreater \ 5$; Range: $y \ \textgreater \ 5$B. Domain: $x \ \textless \ 0$; Range: All Real NumbersC. Domain: $x \ \textgreater \

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Introduction

In mathematics, the domain and range of a function are crucial concepts that help us understand the behavior and properties of the function. The domain of a function is the set of all possible input values, or x-coordinates, that the function can accept, while the range is the set of all possible output values, or y-coordinates, that the function can produce. In this article, we will explore the domain and range of the function f(x)=log⁡x−5f(x) = \log x - 5.

What is the Domain of a Logarithmic Function?

The domain of a logarithmic function is the set of all positive real numbers. This is because the logarithm of a non-positive number is undefined. In other words, the logarithmic function is only defined for values of x that are greater than 0.

What is the Domain of the Function f(x)=log⁡x−5f(x) = \log x - 5?

To determine the domain of the function f(x)=log⁡x−5f(x) = \log x - 5, we need to consider the properties of the logarithmic function. Since the logarithm of a non-positive number is undefined, the domain of the function f(x)=log⁡x−5f(x) = \log x - 5 is also the set of all positive real numbers. However, we need to consider the effect of the constant term -5 on the domain.

The constant term -5 does not affect the domain of the function, as it is simply a vertical shift of the graph of the logarithmic function. Therefore, the domain of the function f(x)=log⁡x−5f(x) = \log x - 5 is still the set of all positive real numbers.

What is the Range of a Logarithmic Function?

The range of a logarithmic function is the set of all real numbers. This is because the logarithmic function can produce any real value, regardless of the input value.

What is the Range of the Function f(x)=log⁡x−5f(x) = \log x - 5?

To determine the range of the function f(x)=log⁡x−5f(x) = \log x - 5, we need to consider the properties of the logarithmic function. Since the logarithmic function can produce any real value, the range of the function f(x)=log⁡x−5f(x) = \log x - 5 is also the set of all real numbers.

However, we need to consider the effect of the constant term -5 on the range. The constant term -5 shifts the graph of the logarithmic function downward by 5 units, which means that the range of the function f(x)=log⁡x−5f(x) = \log x - 5 is the set of all real numbers less than -5.

Conclusion

In conclusion, the domain of the function f(x)=log⁡x−5f(x) = \log x - 5 is the set of all positive real numbers, and the range is the set of all real numbers less than -5. This is because the logarithmic function is only defined for positive real numbers, and the constant term -5 shifts the graph of the logarithmic function downward by 5 units.

Answer

The correct answer is:

A. Domain: x \textgreater 0x \ \textgreater \ 0; Range: y \textless −5y \ \textless \ -5

References

  • [1] "Logarithmic Functions" by Math Open Reference
  • [2] "Domain and Range of a Function" by Khan Academy
    Understanding the Domain and Range of a Logarithmic Function: Q&A ================================================================

Introduction

In our previous article, we explored the domain and range of the function f(x)=log⁡x−5f(x) = \log x - 5. We discussed how the domain of a logarithmic function is the set of all positive real numbers, and how the range is the set of all real numbers. In this article, we will answer some frequently asked questions about the domain and range of a logarithmic function.

Q: What is the domain of a logarithmic function?

A: The domain of a logarithmic function is the set of all positive real numbers. This is because the logarithm of a non-positive number is undefined.

Q: What is the range of a logarithmic function?

A: The range of a logarithmic function is the set of all real numbers. This is because the logarithmic function can produce any real value, regardless of the input value.

Q: How does the constant term affect the domain and range of a logarithmic function?

A: The constant term does not affect the domain of a logarithmic function, but it can affect the range. A constant term shifts the graph of the logarithmic function vertically, which means that the range of the function is shifted downward or upward by the constant term.

Q: What is the domain and range of the function f(x)=log⁡x−5f(x) = \log x - 5?

A: The domain of the function f(x)=log⁡x−5f(x) = \log x - 5 is the set of all positive real numbers, and the range is the set of all real numbers less than -5.

Q: Can a logarithmic function have a domain that includes zero?

A: No, a logarithmic function cannot have a domain that includes zero. This is because the logarithm of zero is undefined.

Q: Can a logarithmic function have a range that includes zero?

A: Yes, a logarithmic function can have a range that includes zero. This is because the logarithmic function can produce any real value, including zero.

Q: How do you determine the domain and range of a logarithmic function?

A: To determine the domain and range of a logarithmic function, you need to consider the properties of the logarithmic function. You need to determine the set of all positive real numbers that the function can accept as input values, and the set of all real numbers that the function can produce as output values.

Q: What are some common mistakes to avoid when determining the domain and range of a logarithmic function?

A: Some common mistakes to avoid when determining the domain and range of a logarithmic function include:

  • Assuming that the domain of a logarithmic function includes zero
  • Assuming that the range of a logarithmic function includes zero
  • Failing to consider the effect of a constant term on the domain and range of a logarithmic function

Conclusion

In conclusion, the domain and range of a logarithmic function are crucial concepts that help us understand the behavior and properties of the function. By understanding the domain and range of a logarithmic function, we can determine the set of all possible input values and output values that the function can accept and produce.

Answer Key

  • Q1: The domain of a logarithmic function is the set of all positive real numbers.
  • Q2: The range of a logarithmic function is the set of all real numbers.
  • Q3: The constant term does not affect the domain of a logarithmic function, but it can affect the range.
  • Q4: The domain of the function f(x)=log⁡x−5f(x) = \log x - 5 is the set of all positive real numbers, and the range is the set of all real numbers less than -5.
  • Q5: No, a logarithmic function cannot have a domain that includes zero.
  • Q6: Yes, a logarithmic function can have a range that includes zero.
  • Q7: To determine the domain and range of a logarithmic function, you need to consider the properties of the logarithmic function.
  • Q8: Some common mistakes to avoid when determining the domain and range of a logarithmic function include assuming that the domain includes zero, assuming that the range includes zero, and failing to consider the effect of a constant term on the domain and range of a logarithmic function.