What is the inverse of the logarithmic function [tex]f(x)=\log _9 x[/tex]?
A. [tex]f^{-1}(x)=x^9[/tex]
B. [tex]f^{-1}(x)=-\log _9 x[/tex]
C. [tex]f^{-1}(x)=9^x[/tex]
D. [tex]f^{-1}(x)=\frac{1}{\log _9 x}[/tex]
Answer (2)
Let y = f ( x ) = lo g 9 x .
Rewrite in exponential form: x = 9 y .
Swap x and y : y = 9 x .
The inverse function is 9 x .
Explanation
Understanding the problem Let's analyze the given function and the possible options for its inverse.
The function is f ( x ) = lo g 9 x . We need to find its inverse, denoted as f − 1 ( x ) . The inverse function essentially reverses the operation of the original function.
Finding the inverse To find the inverse, we can follow these steps:
Let y = f ( x ) . So, y = lo g 9 x .
Rewrite the equation in exponential form. The logarithmic equation y = lo g 9 x is equivalent to the exponential equation x = 9 y .
Swap x and y to find the inverse function. This gives us y = 9 x .
Therefore, the inverse function is f − 1 ( x ) = 9 x .
Comparing with the options Now, let's compare our result with the given options:
f − 1 ( x ) = x 9 - Incorrect.
f − 1 ( x ) = − lo g 9 x - Incorrect.
f − 1 ( x ) = 9 x - Correct.
f − 1 ( x ) = l o g 9 x 1 - Incorrect.
Final Answer Therefore, the inverse of the function f ( x ) = lo g 9 x is f − 1 ( x ) = 9 x .
Examples
Logarithmic functions and their inverses are used extensively in various fields. For instance, in finance, they help calculate the time it takes for an investment to reach a certain value with compound interest. In computer science, logarithms are used to analyze the efficiency of algorithms. The inverse relationship between exponential and logarithmic functions is also crucial in solving exponential growth and decay problems, such as population growth or radioactive decay. Understanding inverse functions allows us to reverse-engineer processes and solve for the initial conditions or rates of change.
The inverse of the logarithmic function f ( x ) = lo g 9 x is given by f − 1 ( x ) = 9 x . This result is derived from rewriting the logarithmic function in exponential form and then switching the variables. The correct choice from the options provided is C: f − 1 ( x ) = 9 x .
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