Consider The Graph Of The Function F ( X ) = Log ⁡ 4 ( X − 2 F(x)=\log _4(x-2 F ( X ) = Lo G 4 ( X − 2 ].Over What Interval Of The Domain Is The Function F F F Positive?A. ( − ∞ , 2 (-\infty, 2 ( − ∞ , 2 ]B. ( − ∞ , 3 (-\infty, 3 ( − ∞ , 3 ]C. ( 3 , ∞ (3, \infty ( 3 , ∞ ]D. ( 2 , ∞ (2, \infty ( 2 , ∞ ]

Mar 12, 2025 by ADMIN 306 views

Understanding the Domain of a Logarithmic Function

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Introduction

Logarithmic functions are a fundamental concept in mathematics, and understanding their behavior is crucial for solving various mathematical problems. In this article, we will focus on the function $f(x)=\log _4(x-2)$ and determine the interval of its domain where the function is positive.

The Nature of Logarithmic Functions

A logarithmic function is defined as the inverse of an exponential function. In other words, if $y = a^x$ , then $x = \log_a(y)$ . The logarithmic function $f(x)=\log _4(x-2)$ is a special case where the base is 4, and the argument is $(x-2)$ .

The Domain of a Logarithmic Function

The domain of a logarithmic function is the set of all possible input values for which the function is defined. In the case of $f(x)=\log _4(x-2)$ , the argument $(x-2)$ must be greater than 0, since the logarithm of a non-positive number is undefined.

Finding the Interval of the Domain

To find the interval of the domain where the function $f$ is positive, we need to determine the values of $x$ for which $(x-2) > 0$ . This inequality can be solved by adding 2 to both sides, resulting in $x > 2$ .

Analyzing the Options

Now that we have determined the interval of the domain where the function $f$ is positive, we can analyze the given options:

A. $(-\infty, 2)$ : This interval is not correct, since the function is not defined for $x \leq 2$ .
B. $(-\infty, 3)$ : This interval is also not correct, since the function is not defined for $x \leq 2$ .
C. $(3, \infty)$ : This interval is not correct, since the function is positive for $x > 2$ , not $x > 3$ .
D. $(2, \infty)$ : This interval is correct, since the function is positive for $x > 2$ .

Conclusion

In conclusion, the function $f(x)=\log _4(x-2)$ is positive over the interval $(2, \infty)$ . This means that for any value of $x$ greater than 2, the function $f$ will be positive.

Final Answer

The final answer is $\boxed{(2, \infty)}$ .

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Introduction

In our previous article, we discussed the function $f(x)=\log _4(x-2)$ and determined the interval of its domain where the function is positive. In this article, we will address some frequently asked questions (FAQs) about logarithmic functions.

Q&A

Q1: What is the domain of a logarithmic function?

A1: The domain of a logarithmic function is the set of all possible input values for which the function is defined. In the case of $f(x)=\log _4(x-2)$ , the argument $(x-2)$ must be greater than 0, since the logarithm of a non-positive number is undefined.

Q2: How do I determine the interval of the domain where the function is positive?

A2: To determine the interval of the domain where the function is positive, you need to solve the inequality $(x-2) > 0$ . This can be done by adding 2 to both sides, resulting in $x > 2$ .

Q3: What is the difference between a logarithmic function and an exponential function?

A3: A logarithmic function is the inverse of an exponential function. In other words, if $y = a^x$ , then $x = \log_a(y)$ . The logarithmic function $f(x)=\log _4(x-2)$ is a special case where the base is 4, and the argument is $(x-2)$ .

Q4: Can a logarithmic function have a negative value?

A4: Yes, a logarithmic function can have a negative value. However, the function $f(x)=\log _4(x-2)$ is only defined for $x > 2$ , and its values are always positive.

Q5: How do I graph a logarithmic function?

A5: To graph a logarithmic function, you can use a graphing calculator or software. Alternatively, you can plot points on a coordinate plane and draw a smooth curve through them.

Q6: What is the relationship between logarithmic and exponential functions?

A6: Logarithmic and exponential functions are inverses of each other. This means that if $y = a^x$ , then $x = \log_a(y)$ .

Q7: Can a logarithmic function have a fractional exponent?

A7: Yes, a logarithmic function can have a fractional exponent. However, the function $f(x)=\log _4(x-2)$ has a base of 4, which is an integer.

Q8: How do I evaluate a logarithmic expression?

A8: To evaluate a logarithmic expression, you need to use the properties of logarithms. For example, $\log_a(b^c) = c \log_a(b)$ .

Q9: What is the difference between a common logarithm and a natural logarithm?

A9: A common logarithm is a logarithm with a base of 10, while a natural logarithm is a logarithm with a base of $e$ .

Q10: Can a logarithmic function be used to model real-world phenomena?

A10: Yes, logarithmic functions can be used to model real-world phenomena, such as population growth, chemical reactions, and financial transactions.

Conclusion

In conclusion, logarithmic functions are an important concept in mathematics, and understanding their behavior is crucial for solving various mathematical problems. We hope that this FAQ article has provided you with a better understanding of logarithmic functions and their applications.

Final Answer

The final answer is $\boxed{(2, \infty)}$ .