What are the domain and range of [tex]f(x)=\log x-5[/tex]?
A. domain: [tex]x\ \textgreater \ 0[/tex]; range: all real numbers
B. domain: [tex]x\ \textless \ 0[/tex]; range: all real numbers
C. domain: [tex]x\ \textgreater \ 5[/tex]; range: [tex]y\ \textgreater \ 5[/tex]
D. domain: [tex]x\ \textgreater \ 5[/tex]; range: [tex]y\ \textgreater \ -5[/tex]
Answer (2)
The domain of f ( x ) = lo g x − 5 is determined by the logarithm, which is only defined for 0"> x > 0 .
The range of the logarithm function is all real numbers.
Subtracting a constant from the logarithm does not change the range.
Therefore, the domain is 0"> x > 0 and the range is all real numbers, so the answer is 0; \text{ range: all real numbers}}"> domain: x > 0 ; range: all real numbers .
Explanation
Understanding Domain and Range We want to determine the domain and range of the function f ( x ) = lo g x − 5 . The domain is the set of all possible input values (x-values) for which the function is defined. The range is the set of all possible output values (y-values) that the function can produce.
Determining the Domain The logarithm function, lo g x , is only defined for positive values of x . This means that the argument of the logarithm must be greater than zero. Therefore, the domain of f ( x ) = lo g x − 5 is all 0"> x > 0 .
Determining the Range The range of the logarithm function lo g x is all real numbers. This means that lo g x can take any real value. Subtracting a constant from a function shifts the graph vertically but does not change the range. Therefore, the range of f ( x ) = lo g x − 5 is also all real numbers.
Final Answer Therefore, the domain of f ( x ) = lo g x − 5 is 0"> x > 0 , and the range is all real numbers.
Examples
Logarithmic functions are used in many real-world applications, such as measuring the intensity of earthquakes on the Richter scale, modeling population growth, and calculating the pH of a solution. Understanding the domain and range of logarithmic functions is crucial for interpreting these applications correctly. For example, in the Richter scale, the magnitude of an earthquake is given by M = lo g 10 ( I / S ) , where I is the intensity of the earthquake and S is the intensity of a standard earthquake. Since the intensity I must be positive, the domain of this function is 0"> I > 0 .
The domain of the function f ( x ) = lo g x − 5 is 0"> x > 0 , and the range is all real numbers. Therefore, the correct answer is option A. The logarithmic function is only defined for positive x-values, and it can output any real number regardless of the constant subtracted.
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