Solve 9^x = 81
Exponential equation, worked out line by line the way a teacher would write it.
Answer
| Solution | x = 2 |
Step-by-step solution
4 steps-
1 Given9^{x} = 81
Solve for x.
-
2 Write the right side as a power of 9
Powers with different bases are hard to compare. Facts like 8 = 2³ and 1/4 = 2⁻² rewrite every power with the same base 9; then the exponents can be compared, or the power replaced by a single unknown.
9^{x} = 9^{2} -
3 Same base on both sides, so the exponents must be equal
An exponential function with a positive base other than 1 never takes the same value twice, so bᵐ = bⁿ can only be true when m = n.
x = 2 -
4 Check\begin{aligned}x = 2:\quad 81 = 81\quad\checkmark\end{aligned}
Substituting each solution back makes both sides equal.
Check by substitution
| x | Left side | Right side | |
|---|---|---|---|
| 2 | 81 | 81 | ✓ |
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