What is the inverse of the logarithmic function [tex]f(x)=\log _2 x[/tex] ?
A. [tex]f^{-1}(x)=x^2[/tex]
B. [tex]f^{-1}(x)=2^x[/tex]
C. [tex]f^{-1}(x)=\log _{x^2}[/tex]
D. [tex]f^{-1}(x)=\frac{1}{\log _2 x}[/tex]
Answer (1)
Write the function as y = lo g 2 x .
Express x in terms of y : x = 2 y .
Swap x and y to find the inverse: y = 2 x .
The inverse function is f − 1 ( x ) = 2 x .
Explanation
Understanding the Problem We are given the function f ( x ) = lo g 2 x and we want to find its inverse, denoted as f − 1 ( x ) . The inverse function essentially reverses the operation of the original function.
Writing the Equation To find the inverse, we start by writing the function as an equation: y = lo g 2 x
Expressing x in terms of y Next, we want to express x in terms of y . By the definition of logarithms, the equation y = lo g 2 x is equivalent to: x = 2 y
Swapping x and y Finally, to find the inverse function, we swap x and y : y = 2 x This gives us the inverse function: f − 1 ( x ) = 2 x
Final Answer Therefore, the inverse of the logarithmic function f ( x ) = lo g 2 x is f − 1 ( x ) = 2 x .
Examples
Logarithmic functions and their inverses are used extensively in computer science, particularly in the analysis of algorithms. For example, the number of times you can divide a problem in half (like in binary search) is related to the base-2 logarithm. The inverse, an exponential function, describes growth rates, such as the number of transistors on a computer chip over time (Moore's Law). Understanding these relationships helps in predicting performance and scaling of systems.
