Verified Answer :

  • Rewrite 27 as \(3^3\) to express both sides of the equation with base 3: \(3^{4x} = (3^3)^{x-3}\).
  • Use the power rule \((a^m)^n = a^{m \cdot n}\) to simplify the exponents: \(3^{4x} = 3^{3x-9}\).
  • Equate the exponents to form the linear equation: \(4x = 3x - 9\).
  • Solve the linear equation to find the value of \(x\): \(x = -9\).
  • Final Answer: \(\boxed{-9}\)

Explanation

  1. Expressing with a Common Base

The equation given is \(3^{4x} = 27^{x-3}\). To solve for \(x\), the first step is to express both sides of the equation with a common base. Since both 3 and 27 are powers of 3, we can rewrite 27 as \(3^3\). Substituting this into the equation, we get:
\[3^{4x} = (3^3)^{x-3}\]

  1. Simplifying the Right Side

Now, apply the power of a power property, which states that \((a^m)^n = a^{m \cdot n}\). Applying this to the right side of the equation, we have:
\[(3^3)^{x-3} = 3^{3(x-3)} = 3^{3x-9}\]
So the equation becomes:
\[3^{4x} = 3^{3x-9}\]

  1. Solving for x

Since the bases are identical (both are 3), we can equate the exponents:
\[4x = 3x - 9\]
To isolate \(x\), subtract \(3x\) from both sides of the equation:
\[4x - 3x = -9\]
\[x = -9\]

  1. Final Conclusion

The calculated value of \(x\) is \(-9\). Checking against the given options, this corresponds to option c. Therefore, the solution is \(\boxed{-9}\).

Examples

Exponential equations like the one solved here are fundamental in various real-life scenarios, such as calculating compound interest or modeling population growth. For instance, if you are looking at how a population grows over time, you might encounter equations where the base represents the growth rate and the exponents represent time increments. Solving these ensures you can predict future values accurately in financial or scientific models.

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