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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]
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