At Westonci.ca, we connect you with the best answers from a community of experienced and knowledgeable individuals. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
To determine the domain and range of the logarithmic function [tex]\( f(x) = \log_7(x) \)[/tex], we'll analyze the properties of logarithmic functions and use the inverse function for justification.
### Domain of [tex]\( f(x) = \log_7(x) \)[/tex]
1. Definition and Properties:
- A logarithmic function [tex]\( \log_b(x) \)[/tex] (where [tex]\( b \)[/tex] is the base and [tex]\( b > 0 \)[/tex], [tex]\( b \neq 1 \)[/tex]) is defined only for positive real numbers. This is because the logarithm represents the power to which the base must be raised to get a certain number, and raising a positive number to any real power cannot result in a non-positive number.
2. Domain:
- Therefore, for [tex]\( f(x) = \log_7(x) \)[/tex], [tex]\( x \)[/tex] must be greater than 0. In other words,
[tex]\[ \text{Domain} \: D = (0, \infty) \][/tex]
### Range of [tex]\( f(x) = \log_7(x) \)[/tex]
1. Understanding the Logarithmic Curve:
- The logarithmic function [tex]\( \log_7(x) \)[/tex] can take any real value from negative infinity to positive infinity as [tex]\( x \)[/tex] moves from just above 0 to positive infinity.
2. Range:
- Therefore,
[tex]\[ \text{Range} \: R = (-\infty, \infty) \][/tex]
### Justification Using the Inverse Function
To justify the domain and range using the inverse function, remember that the inverse of [tex]\( f(x) = \log_7(x) \)[/tex] is the exponential function [tex]\( g(y) = 7^y \)[/tex]:
1. Inverse Function:
- The inverse function of [tex]\( f(x) = \log_7(x) \)[/tex] is given by [tex]\( g(y) = 7^y \)[/tex].
2. Domain and Range of the Inverse Function:
- For [tex]\( g(y) = 7^y \)[/tex]:
- The domain is all real numbers [tex]\( y \)[/tex], i.e., [tex]\( (-\infty, \infty) \)[/tex], because any real number can be used as an exponent.
- The range is [tex]\( (0, \infty) \)[/tex], since raising 7 (a positive number) to any real power results in a positive number greater than 0.
3. Connecting with the Original Function:
- The domain of the original function [tex]\( f(x) = \log_7(x) \)[/tex] should match the range of its inverse [tex]\( g(y) = 7^y \)[/tex], which is [tex]\( (0, \infty) \)[/tex].
- The range of the original function [tex]\( f(x) = \log_7(x) \)[/tex] should match the domain of its inverse [tex]\( g(y) = 7^y \)[/tex], which is [tex]\( (-\infty, \infty) \)[/tex].
Thus, based on this analysis, the domain and range of [tex]\( f(x) = \log_7(x) \)[/tex] are:
- Domain: [tex]\( (0, \infty) \)[/tex]
- Range: [tex]\( (-\infty, \infty) \)[/tex]
### Domain of [tex]\( f(x) = \log_7(x) \)[/tex]
1. Definition and Properties:
- A logarithmic function [tex]\( \log_b(x) \)[/tex] (where [tex]\( b \)[/tex] is the base and [tex]\( b > 0 \)[/tex], [tex]\( b \neq 1 \)[/tex]) is defined only for positive real numbers. This is because the logarithm represents the power to which the base must be raised to get a certain number, and raising a positive number to any real power cannot result in a non-positive number.
2. Domain:
- Therefore, for [tex]\( f(x) = \log_7(x) \)[/tex], [tex]\( x \)[/tex] must be greater than 0. In other words,
[tex]\[ \text{Domain} \: D = (0, \infty) \][/tex]
### Range of [tex]\( f(x) = \log_7(x) \)[/tex]
1. Understanding the Logarithmic Curve:
- The logarithmic function [tex]\( \log_7(x) \)[/tex] can take any real value from negative infinity to positive infinity as [tex]\( x \)[/tex] moves from just above 0 to positive infinity.
2. Range:
- Therefore,
[tex]\[ \text{Range} \: R = (-\infty, \infty) \][/tex]
### Justification Using the Inverse Function
To justify the domain and range using the inverse function, remember that the inverse of [tex]\( f(x) = \log_7(x) \)[/tex] is the exponential function [tex]\( g(y) = 7^y \)[/tex]:
1. Inverse Function:
- The inverse function of [tex]\( f(x) = \log_7(x) \)[/tex] is given by [tex]\( g(y) = 7^y \)[/tex].
2. Domain and Range of the Inverse Function:
- For [tex]\( g(y) = 7^y \)[/tex]:
- The domain is all real numbers [tex]\( y \)[/tex], i.e., [tex]\( (-\infty, \infty) \)[/tex], because any real number can be used as an exponent.
- The range is [tex]\( (0, \infty) \)[/tex], since raising 7 (a positive number) to any real power results in a positive number greater than 0.
3. Connecting with the Original Function:
- The domain of the original function [tex]\( f(x) = \log_7(x) \)[/tex] should match the range of its inverse [tex]\( g(y) = 7^y \)[/tex], which is [tex]\( (0, \infty) \)[/tex].
- The range of the original function [tex]\( f(x) = \log_7(x) \)[/tex] should match the domain of its inverse [tex]\( g(y) = 7^y \)[/tex], which is [tex]\( (-\infty, \infty) \)[/tex].
Thus, based on this analysis, the domain and range of [tex]\( f(x) = \log_7(x) \)[/tex] are:
- Domain: [tex]\( (0, \infty) \)[/tex]
- Range: [tex]\( (-\infty, \infty) \)[/tex]
We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.
