What are the domain and range of [tex]f(x)=\log x-5[/tex]?
A. domain: [tex]x>0[/tex]; range: all real numbers
B. domain: [tex]x<0[/tex]; range: all real numbers
C. domain: [tex]x>5[/tex]; range: [tex]y>5[/tex]
D. domain: [tex]x>5[/tex]; range: [tex]y>-5[/tex]
Answer (2)
The domain of f ( x ) = lo g x − 5 is 0"> x > 0 , and the range is all real numbers. Thus, the correct answer is A: domain: 0"> x > 0 ; range: all real numbers.
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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 f ( x ) = lo g x − 5 is all real numbers, as the logarithm function can take any real value.
Therefore, the domain is 0"> x > 0 and the range is all real numbers.
The final answer is domain: 0"> x > 0 ; range: all real numbers. 0; \text{ range: all real numbers}}}"> domain: x > 0 ; range: all real numbers
Explanation
Understanding the Problem We are asked to find the domain and range of the function f ( x ) = lo g x − 5 . Let's break this down.
Determining the Domain The domain of a function is the set of all possible input values (x-values) for which the function is defined. The logarithm function, lo g x , is only defined for positive values of x . Therefore, the domain of f ( x ) = lo g x − 5 is all 0"> x > 0 .
Determining the Range The range of a function is the set of all possible output values (y-values) that the function can produce. The range of the logarithm function, lo g x , is all real numbers. Subtracting a constant (in this case, 5) from the logarithm function shifts the graph vertically but does not change the range. Therefore, the range of f ( x ) = lo g x − 5 is 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 models and making accurate predictions. For example, if we are modeling the population growth of a bacteria colony using a logarithmic function, the domain tells us the valid time intervals for which the model is applicable, and the range tells us the possible population sizes.
