What are the domain and range of the logarithmic function $f(x)=\log _7 x$ ? Use the inverse function to justify your answers.
Answer (1)
Find the inverse function of f ( x ) = lo g 7 x , which is f − 1 ( x ) = 7 x .
Determine the domain of the inverse function f − 1 ( x ) = 7 x , which is ( − ∞ , ∞ ) .
Determine the range of the inverse function f − 1 ( x ) = 7 x , which is ( 0 , ∞ ) .
The domain of f ( x ) = lo g 7 x is ( 0 , ∞ ) , and the range is ( − ∞ , ∞ ) .
Domain: ( 0 , ∞ ) , Range: ( − ∞ , ∞ )
Explanation
Problem Analysis Let's analyze the logarithmic function f ( x ) = lo g 7 x and determine its domain and range using the inverse function.
Finding the Inverse Function To find the inverse function, we start by writing the function as y = lo g 7 x . Then, we swap x and y to get x = lo g 7 y .
Determining the Inverse Function Now, we solve for y . The equation x = lo g 7 y can be rewritten in exponential form as y = 7 x . Therefore, the inverse function is f − 1 ( x ) = 7 x .
Relating Domain and Range The domain of the original function f ( x ) is the range of its inverse f − 1 ( x ) , and the range of f ( x ) is the domain of f − 1 ( x ) . So, let's find the domain and range of the inverse function f − 1 ( x ) = 7 x .
Domain of the Inverse Function The domain of the exponential function f − 1 ( x ) = 7 x is all real numbers, since we can raise 7 to any power. In interval notation, the domain is ( − ∞ , ∞ ) .
Range of the Inverse Function The range of the exponential function f − 1 ( x ) = 7 x is all positive real numbers, since 7 x is always positive for any real number x . In interval notation, the range is ( 0 , ∞ ) .
Domain and Range of the Original Function Therefore, the domain of the original function f ( x ) = lo g 7 x is the range of the inverse function, which is ( 0 , ∞ ) . The range of the original function f ( x ) = lo g 7 x is the domain of the inverse function, which is ( − ∞ , ∞ ) .
Final Answer In conclusion, the domain of f ( x ) = lo g 7 x is ( 0 , ∞ ) , and the range is ( − ∞ , ∞ ) .
Examples
Logarithmic functions are incredibly useful in many real-world scenarios. For instance, the Richter scale, which measures the magnitude of earthquakes, is a logarithmic scale. An earthquake of magnitude 7 is ten times more powerful than an earthquake of magnitude 6. Similarly, the decibel scale, used to measure sound intensity, is also logarithmic. These scales allow us to represent very large ranges of values in a more manageable way. Understanding the domain and range of logarithmic functions helps us interpret these scales correctly and make informed decisions based on the data they provide.
