Equivalent Exponential Equations: (1/3)^x = 27^(x+2)

Let's dive into the world of exponential equations and figure out which option is equivalent to the given equation:

Understanding the Core Concept

When we're faced with an equation like $\left(\frac{1}{3}\right)^x=27^{x+2}$, the key to solving it and finding an equivalent form lies in expressing both sides of the equation with the same base. Remember, if we have an equation of the form $a^m = a^n$, then it must be true that $m=n$. So, our primary goal is to manipulate the given equation until both sides have a common base.

Step-by-Step Solution

1. Analyze the Bases: We have a base of $\frac{1}{3}$ on the left side and a base of $27$ on the right side. Neither of these are the same, so we need to find a common base that can be used to express both $\frac{1}{3}$ and $27$.

2. Identify a Common Base: Let's think about the relationship between $3$, $\frac{1}{3}$, and $27$. We know that $3^1 = 3$. We also know that $\frac{1}{3}$ can be written as $3^{-1}$ using the property of negative exponents ($a^{-n} = \frac{1}{a^n}$). And $27$ is simply $3 \times 3 \times 3$, which is $3^3$. Aha! The number $3$ is our common base.

3. Rewrite the Left Side: Let's substitute $3^{-1}$ for $\frac{1}{3}$ in the original equation. So, $\left(\frac{1}{3}\right)^x$ becomes $\left(3^{-1}\right)^x$. Using the power of a power rule ($(a^m)^n = a^{mn}$), this simplifies to $3^{-1 \times x}$, which is $3^{-x}$.

4. Rewrite the Right Side: Now, let's work on the right side, $27^{x+2}$. We know that $27 = 3^3$. So, we can rewrite $27^{x+2}$ as $\left(3^3\right)^{x+2}$. Again, applying the power of a power rule, we multiply the exponents: $3^{3 \times (x+2)}$. Distributing the $3$, we get $3^{3x+6}$.

5. Combine and Equate: Now that we have both sides expressed with the same base ($3$), we can rewrite the original equation as: $3^{-x} = 3^{3x+6}$

6. Equate the Exponents: Since the bases are the same, we can now equate the exponents: $-x = 3x+6$

7. Solve for x (Optional, but good for verification): Subtract $3x$ from both sides: $-x - 3x = 6$, which gives $-4x = 6$. Divide by $-4$: $x = \frac{6}{-4}$, which simplifies to $x = -\frac{3}{2}$.

Comparing with the Options

Now, let's look at the given options and see which one matches our derived equation $3^{-x} = 3^{3x+6}$.

• A. $3^x=3^{-3 x+2}$: This does not match our derived equation.
• B. $3^x=3^{3 x+6}$: This does not match our derived equation.
• C. $3^{-x}=3^{3 x+2}$: This has the correct left side, but the exponent on the right side is incorrect.
• D. $3^{-x}=3^{3 x+6}$: This perfectly matches our derived equation!

Conclusion

By carefully rewriting both sides of the original equation $\left(\frac{1}{3}\right)^x=27^{x+2}$ with a common base of $3$, we arrived at the equivalent equation $3^{-x} = 3^{3x+6}$. Therefore, option D is the correct answer.

This process highlights the importance of understanding exponent rules, such as negative exponents and the power of a power rule, when working with exponential equations. Mastering these rules allows us to transform complex-looking equations into simpler, more manageable forms.

For further exploration into exponential functions and their properties, you can check out resources like Khan Academy's section on exponential functions. They offer a comprehensive overview and practice problems to solidify your understanding.