To solve the equation \( 3^{2x} = 81 \), we need to find the value of \( x \). Let's break this down step-by-step. ### Step 1: Rewrite 81 as a power of 3 First, we need to express 81 in terms of a base of 3. We know that: \[ 81 = 3^4 \] This means we can rewrite the original equation as: \[ 3^{2x} = 3^4 \] ### Step 2: Set the exponents equal to each other Since the bases are the same (both are base 3), we can set the exponents equal to each other: \[ 2x = 4 \] ### Step 3: Solve for \( x \) Now, we can solve for \( x \) by dividing both sides of the equation by 2: \[ x = \frac{4}{2} = 2 \] ### Conclusion Thus, the value of \( x \) is \( 2 \). Therefore, the correct option is **B**. ### Explanation of Other Options - **Option A (1)**: If \( x = 1 \), then \( 3^{2x} = 3^{2 \cdot 1} = 3^2 = 9 \), which is not equal to 81. - **Option C (3)**: If \( x = 3 \), then \( 3^{2x} = 3^{2 \cdot 3} = 3^6 = 729 \), which is also not equal to 81. - **Option D (4)**: If \( x = 4 \), then \( 3^{2x} = 3^{2 \cdot 4} = 3^8 = 6561 \), which is far greater than 81. ### Common Pitfalls - **Not recognizing powers**: A common mistake is not rewriting numbers as powers of the same base. Always check if you can express both sides of the equation with the same base. - **Forgetting to set exponents equal**: When the bases are the same, it’s crucial to remember that you can equate the exponents directly. ### Revision Summary - To solve exponential equations, express both sides with the same base if possible. - Set the exponents equal to each other when the bases are the same. - Solve for the variable by isolating it. - Always check your solution by substituting it back into the original equation. By following these steps, you can confidently tackle similar problems in the future!