What Is The Inverse Of The Logarithmic Function $f(x) = \log_9 X$?A. $f^{-1}(x) = X^9$B. $f^{-1}(x) = -\log_9 X$C. $f^{-1}(x) = 9^x$D. $f^{-1}(x) = \frac{1}{\log_9 X}$

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Introduction

In mathematics, the concept of inverse functions is crucial in understanding the relationship between different functions. The inverse of a function essentially reverses the operation of the original function, allowing us to find the input value that corresponds to a given output value. In this article, we will explore the inverse of the logarithmic function f(x)=log⁡9xf(x) = \log_9 x and determine the correct answer among the given options.

Understanding Logarithmic Functions

Before we dive into finding the inverse of the logarithmic function, let's briefly review what logarithmic functions are. A logarithmic function is a function that takes a positive real number as input and returns the exponent to which a fixed base must be raised to produce the input value. In other words, if y=log⁡bxy = \log_b x, then by=xb^y = x. The base of the logarithm is a fixed number, and the exponent is the input value.

Finding the Inverse of the Logarithmic Function

To find the inverse of the logarithmic function f(x)=log⁡9xf(x) = \log_9 x, we need to reverse the operation of the function. This means that we need to find the input value that corresponds to a given output value. Let's denote the inverse function as f−1(x)f^{-1}(x). We want to find the expression for f−1(x)f^{-1}(x) in terms of xx.

Using the Definition of Inverse Functions

By definition, the inverse function f−1(x)f^{-1}(x) satisfies the following property:

f(f−1(x))=xf(f^{-1}(x)) = x

Substituting f(x)=log⁡9xf(x) = \log_9 x into this equation, we get:

log⁡9(log⁡9−1x)=x\log_9 (\log_9^{-1} x) = x

Simplifying the Equation

To simplify the equation, we can use the fact that log⁡ba=c\log_b a = c is equivalent to bc=ab^c = a. Applying this to the equation above, we get:

9log⁡9−1x=x9^{\log_9^{-1} x} = x

Solving for the Inverse Function

Now, we need to solve for log⁡9−1x\log_9^{-1} x. To do this, we can take the logarithm base 9 of both sides of the equation:

log⁡9(9log⁡9−1x)=log⁡9x\log_9 (9^{\log_9^{-1} x}) = \log_9 x

Using the property of logarithms that log⁡b(bc)=c\log_b (b^c) = c, we get:

log⁡9−1x=log⁡9x\log_9^{-1} x = \log_9 x

Conclusion

Therefore, the inverse of the logarithmic function f(x)=log⁡9xf(x) = \log_9 x is f−1(x)=log⁡9xf^{-1}(x) = \log_9 x. This means that the correct answer among the given options is:

C. f−1(x)=9xf^{-1}(x) = 9^x

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Q&A

Q: What is the inverse of a function?

A: The inverse of a function is a function that reverses the operation of the original function. In other words, if f(x)f(x) is a function, then its inverse f−1(x)f^{-1}(x) is a function that satisfies the property f(f−1(x))=xf(f^{-1}(x)) = x.

Q: How do I find the inverse of a logarithmic function?

A: To find the inverse of a logarithmic function, you need to reverse the operation of the function. This means that you need to find the input value that corresponds to a given output value. Let's denote the inverse function as f−1(x)f^{-1}(x). We want to find the expression for f−1(x)f^{-1}(x) in terms of xx.

Q: What is the inverse of the logarithmic function f(x)=log⁡9xf(x) = \log_9 x?

A: To find the inverse of the logarithmic function f(x)=log⁡9xf(x) = \log_9 x, we need to reverse the operation of the function. This means that we need to find the input value that corresponds to a given output value. Let's denote the inverse function as f−1(x)f^{-1}(x). We want to find the expression for f−1(x)f^{-1}(x) in terms of xx.

Q: How do I simplify the equation to find the inverse function?

A: To simplify the equation, we can use the fact that log⁡ba=c\log_b a = c is equivalent to bc=ab^c = a. Applying this to the equation above, we get:

9log⁡9−1x=x9^{\log_9^{-1} x} = x

Q: What is the final expression for the inverse function?

A: The final expression for the inverse function is:

f−1(x)=log⁡9xf^{-1}(x) = \log_9 x

Q: What is the correct answer among the given options?

A: The correct answer among the given options is:

C. f−1(x)=9xf^{-1}(x) = 9^x

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