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The problem involves finding the inverse of the function \( f(x) = 27^x \). The inverse function, \( f^{-1}(x) \), reverses the effect of \( f(x) \), meaning it solves for \( x \) in terms of \( y \) when \( y = 27^x \).

### Steps to Find the Inverse:
1. Start with the function \( y = 27^x \).
2. To find \( f^{-1}(x) \), switch \( x \) and \( y \), giving \( x = 27^y \).
3. Solve for \( y \) in terms of \( x \). Applying logarithms (preferably the logarithm base 27 for simplicity), we get:
   \[
   \log_{27} x = y
   \]

Thus, \( f^{-1}(x) = \log_{27} x \). Checking the given options, none of them exactly match \(\log_{27} x\), but since \( 27 = 3^3 \), we can express it using a change of base formula:

\[
\log_{27} x = \frac{\log_3 x}{\log_3 27} = \frac{\log_3 x}{3}
\]

### Given Options:
- \( \log_{27} x \): This matches our calculation for the inverse function.
- \( 27x \): Incorrect, this would imply multiplication, not the inverse operation.
- \( 2^{x} \): Incorrect, not related to base 27 logarithm.
- \( \log_2 27 \): Incorrect, this is a constant, not a function of \( x \).

Therefore, the correct answer is \( \log_{27} x \).

Do you have any more questions about this problem, or would you like more details? Here are some related questions you might consider:
1. How can you derive the logarithm change of base formula?
2. What would \( f^{-1}(x) \) be if the base were different, say \( 10 \) instead of \( 27 \)?
3. Why is the exponential function \( 27^x \) one-to-one and therefore invertible?
4. Can you solve for \( x \) if given a specific value for \( f^{-1}(x) \), such as \( f^{-1}(81) \)?
5. What are the properties of logarithmic functions that make them suitable as inverse functions for exponentials?

**Tip:** Remember, the logarithm base you choose can simplify calculations, especially when it matches the base of the exponential function you're working with.

Learn how to find the inverse function of the exponential function f(x) = 27^x. This problem demonstrates the use of logarithms to solve for f^{-1}(x). Suitable for students in Grades 9-12 studying inverse functions and logarithms.

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