Question
Real Tutor Solution
Tutor-Verified Answer
Answer
\( x = \dfrac{4}{5} \)
Solution
To solve the equation \( 243^{x} = 81 \) for \( x \), follow these steps:
1. **Express both numbers as powers of the same base:**
Notice that both 243 and 81 can be written as powers of 3:
\[
243 = 3^5 \quad \text{and} \quad 81 = 3^4
\]
2. **Rewrite the original equation using these expressions:**
\[
(3^5)^x = 3^4
\]
Simplifying the left side using the power of a power rule \((a^m)^n = a^{mn}\):
\[
3^{5x} = 3^4
\]
3. **Set the exponents equal to each other:**
Since the bases are the same and the equation holds true, the exponents must be equal:
\[
5x = 4
\]
4. **Solve for \( x \):**
\[
x = \frac{4}{5}
\]
**Final Answer:**
\( x = \dfrac{4}{5} \)
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Simplify this solution The Deep Dive
To solve the equation \( 243^{x} = 81 \), we can first express both sides with a common base. Notably, both 243 and 81 can be written as powers of 3: \( 243 = 3^5 \) and \( 81 = 3^4 \). Now, we can rewrite the equation as: \[ (3^5)^{x} = 3^4 \] This simplifies to: \[ 3^{5x} = 3^4 \] Since the bases are the same, we can set the exponents equal to each other: \[ 5x = 4 \] Solving for \( x \): \[ x = \frac{4}{5} \] Thus, the solution to the equation \( 243^{x} = 81 \) is \( x = \frac{4}{5} \).
