To solve the equation \( 3^{x} = 81 \), we need to find the value of \( x \) that makes this equation true. Let's break this down step-by-step. ### Step 1: Understand the Equation The equation \( 3^{x} = 81 \) means that \( 3 \) raised to the power of \( x \) equals \( 81 \). Our goal is to express \( 81 \) as a power of \( 3 \) so that we can easily compare the exponents. ### Step 2: Express 81 as a Power of 3 To find out what power of \( 3 \) equals \( 81 \), we can start by calculating the powers of \( 3 \): - \( 3^1 = 3 \) - \( 3^2 = 9 \) - \( 3^3 = 27 \) - \( 3^4 = 81 \) From this calculation, we see that \( 3^4 = 81 \). ### Step 3: Set the Exponents Equal Since we have established that \( 81 = 3^4 \), we can rewrite our original equation: \[ 3^{x} = 3^{4} \] ### Step 4: Solve for \( x \) When the bases are the same (both are \( 3 \)), we can set the exponents equal to each other: \[ x = 4 \] ### Conclusion Thus, the value of \( x \) in the equation \( 3^{x} = 81 \) is \( 4 \). ### Final Answer The correct option is **C. 4**. ### Explanation of Other Options - **A. 2**: This option is incorrect because \( 3^2 = 9 \), which is not equal to \( 81 \). - **B. 3**: This option is also incorrect because \( 3^3 = 27 \), which is still not equal to \( 81 \). - **D. 5**: This option is incorrect as well because \( 3^5 = 243 \), which is greater than \( 81 \). ### Summary of Key Points - To solve \( 3^{x} = 81 \), express \( 81 \) as a power of \( 3 \). - Recognize that \( 81 = 3^4 \). - Set the exponents equal to find \( x = 4 \). - The correct answer is **C. 4**. This method of rewriting numbers as powers of their bases is a common technique in solving exponential equations, and it’s important to practice this skill for future problems.