What Is The Domain Of $y = \log_5 X$?A. All Real Numbers Less Than 0 B. All Real Numbers Greater Than 0 C. All Real Numbers Not Equal To 0 D. All Real Numbers

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Understanding the Concept of Domain

The domain of a function is the set of all possible input values for which the function is defined. In other words, it is the set of all possible values of the variable that the function can accept without resulting in an undefined or imaginary output. In the context of logarithmic functions, the domain is particularly important because the logarithm of a non-positive number is undefined.

The Logarithmic Function

The logarithmic function $y = \log_b x$ is defined as the inverse of the exponential function $y = b^x$. The base $b$ is a positive real number, and the domain of the logarithmic function is all real numbers greater than 0. This is because the logarithm of a non-positive number is undefined, and the exponential function $b^x$ is defined for all real numbers.

The Domain of $y = \log_5 x$

In the case of the function $y = \log_5 x$, the base is 5, which is a positive real number. Therefore, the domain of this function is all real numbers greater than 0. This means that the function is defined for all positive real numbers, but it is undefined for all non-positive real numbers.

Analyzing the Options

Let's analyze the options given in the discussion category:

A. All real numbers less than 0 B. All real numbers greater than 0 C. All real numbers not equal to 0 D. All real numbers

Option A is incorrect because the logarithm of a non-positive number is undefined. Option C is also incorrect because the logarithm of 0 is undefined. Option D is incorrect because the logarithm of a non-positive number is undefined.

Conclusion

The correct answer is option B: All real numbers greater than 0. This is because the domain of the function $y = \log_5 x$ is all real numbers greater than 0, which means that the function is defined for all positive real numbers.

Example Use Case

To illustrate the concept of domain, let's consider an example. Suppose we want to find the domain of the function $y = \log_5 (x^2)$. Since the function is defined for all real numbers greater than 0, the domain of this function is also all real numbers greater than 0.

Key Takeaways

• The domain of a function is the set of all possible input values for which the function is defined.
• The logarithmic function $y = \log_b x$ is defined for all real numbers greater than 0.
• The domain of the function $y = \log_5 x$ is all real numbers greater than 0.

Final Thoughts

In conclusion, the domain of the function $y = \log_5 x$ is all real numbers greater than 0. This is because the logarithm of a non-positive number is undefined, and the exponential function $5^x$ is defined for all real numbers. Understanding the concept of domain is crucial in mathematics, and it has numerous applications in various fields such as science, engineering, and economics.

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Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values for which the function is defined.

Q: Why is the domain of a function important?

A: The domain of a function is important because it determines the set of all possible input values that the function can accept without resulting in an undefined or imaginary output.

Q: What is the domain of the logarithmic function $y = \log_b x$?

A: The domain of the logarithmic function $y = \log_b x$ is all real numbers greater than 0.

Q: Why is the domain of the logarithmic function $y = \log_b x$ all real numbers greater than 0?

A: The domain of the logarithmic function $y = \log_b x$ is all real numbers greater than 0 because the logarithm of a non-positive number is undefined.

Q: What is the domain of the function $y = \log_5 x$?

A: The domain of the function $y = \log_5 x$ is all real numbers greater than 0.

Q: Why is the domain of the function $y = \log_5 x$ all real numbers greater than 0?

A: The domain of the function $y = \log_5 x$ is all real numbers greater than 0 because the logarithm of a non-positive number is undefined.

Q: Can the domain of a function be all real numbers?

A: No, the domain of a function cannot be all real numbers. This is because the logarithm of a non-positive number is undefined.

Q: Can the domain of a function be all real numbers less than 0?

A: No, the domain of a function cannot be all real numbers less than 0. This is because the logarithm of a non-positive number is undefined.

Q: Can the domain of a function be all real numbers not equal to 0?

A: No, the domain of a function cannot be all real numbers not equal to 0. This is because the logarithm of 0 is undefined.

Q: What is the relationship between the domain of a function and its graph?

A: The domain of a function is related to its graph in that the graph of a function is only defined for the values of x that are in the domain of the function.

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

A: To determine the domain of a function, you need to identify the values of x for which the function is defined. This can be done by analyzing the function and identifying any restrictions on the values of x.

Q: What are some common restrictions on the domain of a function?

A: Some common restrictions on the domain of a function include:

• The function is undefined for non-positive values of x.
• The function is undefined for values of x that are not real numbers.
• The function is undefined for values of x that are not in the range of the inverse function.

Q: How do you write the domain of a function in interval notation?

A: To write the domain of a function in interval notation, you need to identify the values of x for which the function is defined and then write them in interval notation.

Q: What is the difference between the domain and the range of a function?

A: The domain of a function is the set of all possible input values for which the function is defined, while the range of a function is the set of all possible output values of the function.

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

A: To determine the range of a function, you need to identify the values of y for which the function is defined. This can be done by analyzing the function and identifying any restrictions on the values of y.

Q: What are some common restrictions on the range of a function?

A: Some common restrictions on the range of a function include:

• The function is undefined for non-real values of y.
• The function is undefined for values of y that are not in the range of the inverse function.

Q: How do you write the range of a function in interval notation?

A: To write the range of a function in interval notation, you need to identify the values of y for which the function is defined and then write them in interval notation.

Q: What is the relationship between the domain and the range of a function?

A: The domain and the range of a function are related in that the domain of a function determines the set of all possible input values for which the function is defined, while the range of a function determines the set of all possible output values of the function.