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)
Let y = f ( x ) = lo g 2 x .
Swap x and y to get x = lo g 2 y .
Rewrite in exponential form: y = 2 x .
The inverse function is f − 1 ( x ) = 2 x .
Explanation
Understanding the Problem The problem asks us to find the inverse of the function f ( x ) = lo g 2 x . To find the inverse of a function, we swap x and y and then solve for y .
Swapping x and y Let y = f ( x ) = lo g 2 x . To find the inverse, we swap x and y to get x = lo g 2 y .
Solving for y Now we solve for y in terms of x . The equation x = lo g 2 y is in logarithmic form. We can rewrite it in exponential form as y = 2 x .
The Inverse Function Therefore, the inverse function is f − 1 ( x ) = 2 x .
Examples
Logarithmic functions and their inverses are used in many scientific and engineering applications. For example, the Richter scale, which measures the magnitude of earthquakes, is a logarithmic scale. To determine the energy released by an earthquake, we need to use the inverse function, which is an exponential function. Similarly, in computer science, logarithms are used to analyze the efficiency of algorithms, and exponential functions are used in cryptography.
