f(x)=\log _{10} x
What is the domain of the function?
What is the range of the function?
Answer (1)
The domain of the logarithmic function f ( x ) = lo g 10 x is all positive real numbers: ( 0 , ∞ ) .
The range of the logarithmic function f ( x ) = lo g 10 x is all real numbers: ( − ∞ , ∞ ) .
The point ( 0.2 , − 1 ) lies on the graph of the function.
Therefore, the domain is ( 0 , ∞ ) and the range is ( − ∞ , ∞ ) .
Explanation
Understanding the Problem The problem asks us to find the domain and range of the function f ( x ) = lo g 10 x . We also know that the point ( 0.2 , − 1 ) lies on the graph of the function.
Finding the Domain The domain of a logarithmic function f ( x ) = lo g b x is the set of all positive real numbers, since we can only take the logarithm of a positive number. In interval notation, this is ( 0 , ∞ ) .
Finding the Range The range of a logarithmic function f ( x ) = lo g b x is the set of all real numbers. This means that the function can take any real number as its output. In interval notation, this is ( − ∞ , ∞ ) .
Final Answer Therefore, the domain of the function f ( x ) = lo g 10 x is ( 0 , ∞ ) and the range is ( − ∞ , ∞ ) .
Examples
Logarithmic functions are used in many real-world applications, such as measuring the intensity of earthquakes on the Richter scale, determining the pH of a solution in chemistry, and modeling population growth in biology. Understanding the domain and range of logarithmic functions is crucial for interpreting these applications correctly. For example, when measuring earthquake intensity, the magnitude cannot be negative, which relates to the domain of the logarithmic scale.
