Questions: Solve the equation for (x). [ left(frac19right)^x=81 ]

$Solve the equation for (x). [ left(frac19right)^x=81 ]$

Transcript text: Solve the equation for $x$. \[ \left(\frac{1}{9}\right)^{x}=81 \]

Solution

Step 1: Express Both Sides with the Same Base

We start with the equation: \[ \left(\frac{1}{9}\right)^{x} = 81 \]

We can express $\frac{1}{9}$ as $9^{-1}$ and $81$ as $9^2$: \[ (9^{-1})^x = 9^2 \]

Step 2: Simplify the Exponents

Using the property of exponents $(a^m)^n = a^{mn}$, we get: \[ 9^{-x} = 9^2 \]

Step 3: Set the Exponents Equal to Each Other

Since the bases are the same, we can set the exponents equal to each other: \[ -x = 2 \]

Step 4: Solve for $x$

Solving for $x$, we get: \[ x = -2 \]

\[ \boxed{x = -2} \]

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