Inverse Function Of Logarithmic Functions: A Step-by-Step Guide

Have you ever wondered how to reverse a logarithmic function? Understanding inverse functions is crucial in mathematics, and today, we're diving deep into finding the inverse of a logarithmic function. Specifically, we'll tackle the function ${ f(x) = \log_2(\log_3 x) }$. By the end of this guide, you’ll not only know the answer but also understand the process, making similar problems a breeze.

Understanding the Problem: Finding the Inverse Function

At its core, finding the inverse function means reversing the roles of ${ x }$ and ${ y }$. If our function takes ${ x }$ as an input and gives us ${ y }$ as an output, the inverse function takes ${ y }$ as an input and gives us the original ${ x }$. This might sound abstract, but we'll break it down step by step. Our main goal here is to find an expression for ${ f^{-1}(x) }$, given that ${ f(x) = \log_2(\log_3 x) }$. This involves a few key steps that we will explore in detail.

The Foundation: Logarithmic Functions and Inverses

Before we jump into the specifics, let's quickly recap logarithmic functions and their inverses. A logarithm is essentially the inverse operation to exponentiation. The expression ${ \log_b a = c }$ means that ${ b^c = a }$. Understanding this relationship is fundamental to unraveling our problem. The inverse of a logarithmic function is an exponential function, and vice versa. This is the cornerstone of our approach to finding ${ f^{-1}(x) }$.

Step-by-Step Solution: Unraveling the Layers

Now, let's get to the heart of the matter. We are given ${ f(x) = \log_2(\log_3 x) }$. To find the inverse, we'll follow these steps:

1. Replace ${ f(x) }$ with ${ y }$: This gives us ${ y = \log_2(\log_3 x) }$.
2. Swap ${ x }$ and ${ y }$: This is the crucial step in finding the inverse, resulting in ${ x = \log_2(\log_3 y) }$.
3. Isolate ${ y }$: This is where the magic happens. We need to undo the logarithmic layers to get ${ y }$ by itself.
   • First, we tackle the outer logarithm. Since ${ x = \log_2(\log_3 y) }$, we can rewrite this in exponential form as ${ 2^x = \log_3 y }$. This step undoes the ${ \log_2 }$ operation.
   • Next, we address the inner logarithm. We have ${ 2^x = \log_3 y }$. Converting this to exponential form gives us ${ 3^{(2^x)} = y }$. Here, we've undone the ${ \log_3 }$ operation.
4. Replace ${ y }$ with ${ f^{-1}(x) }$: Finally, we express our inverse function as ${ f^{-1}(x) = 3^{(2^x)} }$.

Deep Dive into Each Step

Let’s dissect each step to ensure clarity. When we replace ${ f(x) }$ with ${ y }$, it’s a simple notational change, setting the stage for the inverse process. Swapping ${ x }$ and ${ y }$ is the core of finding an inverse; we are literally reversing the function's input and output. The real work begins when we isolate ${ y }$. This involves using the fundamental relationship between logarithms and exponentials. Remember, ${ \log_b a = c }$ is equivalent to ${ b^c = a }$. Applying this twice, we peel away the logarithmic layers, first undoing the base-2 logarithm and then the base-3 logarithm. This meticulous process reveals the inverse function.

Why This Works: The Essence of Inverse Functions

The reason this step-by-step method works is rooted in the very definition of inverse functions. An inverse function undoes the action of the original function. In our case, ${ f(x) }$ first takes ${ x }$, applies a ${ \log_3 }$, and then a ${ \log_2 }$. The inverse function, ${ f^{-1}(x) }$, reverses this process: it takes ${ x }$, uses it as an exponent with base 2, and then uses the result as an exponent with base 3. This perfect reversal is why we can confidently say that ${ f^{-1}(x) = 3^{(2^x)} }$ is indeed the inverse of ${ f(x) = \log_2(\log_3 x) }$.

Verifying the Inverse Function

To be absolutely sure we have the correct inverse, we can verify it. A function and its inverse should “cancel each other out.” Mathematically, this means that ${ f(f^{-1}(x)) = x }$ and ${ f^{-1}(f(x)) = x }$. Let's test this with our functions.

Testing ${ f(f^{-1}(x)) }$

We need to evaluate ${ f(3^{(2^x)}) }$. Recall that ${ f(x) = \log_2(\log_3 x) }$. So, we have:

${ f(3^{(2^x)}) = \log_2(\log_3 (3^{(2^x)})) }$

Using the property of logarithms that ${ \log_b (b^a) = a }$, we simplify the inner logarithm:

${ \log_3 (3^{(2^x)}) = 2^x }$

Now our expression becomes:

${ f(3^{(2^x)}) = \log_2(2^x) }$

Applying the same logarithm property again, we get:

${ \log_2(2^x) = x }$

So, ${ f(f^{-1}(x)) = x }$, which is exactly what we wanted.

Testing ${ f^{-1}(f(x)) }$

Now let's evaluate ${ f^{-1}(f(x)) }$, which means ${ f^{-1}(\log_2(\log_3 x)) }$. Recall that ${ f^{-1}(x) = 3^{(2^x)} }$. Substituting, we have:

${ f^{-1}(\log_2(\log_3 x)) = 3^{(2^{(\log_2(\log_3 x))})} }$

Let's focus on the exponent: ${ 2^{(\log_2(\log_3 x))} }$. Using the property that ${ b^{\log_b a} = a }$, this simplifies to:

${ 2^{(\log_2(\log_3 x))} = \log_3 x }$

Now our expression becomes:

${ f^{-1}(\log_2(\log_3 x)) = 3^{(\log_3 x)} }$

Applying the property ${ b^{\log_b a} = a }$ one more time, we get:

${ 3^{(\log_3 x)} = x }$

Thus, ${ f^{-1}(f(x)) = x }$, confirming our inverse function.

Why Verification Matters

Verifying the inverse function is a critical step. It ensures that we haven't made any errors in our calculations. By showing that both ${ f(f^{-1}(x)) = x }$ and ${ f^{-1}(f(x)) = x }$, we provide a rigorous confirmation of our result. This process solidifies our understanding and gives us confidence in our solution.

Conclusion: Mastering Inverse Logarithmic Functions

In conclusion, finding the inverse of the function ${ f(x) = \log_2(\log_3 x) }$ involves a methodical process of swapping variables and undoing logarithmic layers using exponential properties. We successfully found that ${ f^{-1}(x) = 3^{(2^x)} }$ and verified this result by showing that the composition of the function and its inverse yields ${ x }$. This journey underscores the importance of understanding the relationship between logarithmic and exponential functions and the fundamental concept of inverse functions.

By mastering these techniques, you'll be well-equipped to tackle a wide range of mathematical problems involving logarithmic and exponential functions. Keep practicing, and you'll find these concepts becoming second nature!

For further exploration and to deepen your understanding of inverse functions, visit Khan Academy's section on Inverse Functions.