What are the domain and range of $f(x)=\log x-5$?
A. domain: $x>0$; range: all real numbers
B. domain: $x<0$; range: all real numbers
C. domain: $x>5$; range: $y>5$
D. domain: $x>5$; range: $y>-5$
Answer (1)
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 the same as the range of lo g x , which is all real numbers.
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.
Explanation
Understanding the Problem The function given is f ( x ) = lo g x − 5 . We need to find the domain and range of this function.
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 or modeling population growth. Understanding the domain and range of logarithmic functions is crucial for interpreting these models correctly. For example, if we are modeling the population growth of a bacteria colony using a logarithmic function, the domain tells us the time interval for which the model is valid, and the range tells us the possible population sizes.
