Questions: Solve for (x) : [ log 3left(x^9right)=-2 ] (x=) (square)

Transcript text: Solve for $x$ : \[ \log _{3}\left(x^{9}\right)=-2 \] $x=$ $\square$

Solution

Solution Steps

To solve the logarithmic equation $\log_{3}(x^9) = -2$, we can convert it to its exponential form. The equation $\log_{b}(a) = c$ is equivalent to $b^c = a$. Applying this to the given equation, we get $3^{-2} = x^9$. We then solve for $x$ by taking the ninth root of both sides.

Step 1: Convert to Exponential Form

We start with the logarithmic equation: \[ \log_{3}(x^9) = -2 \] Converting this to exponential form gives us: \[ 3^{-2} = x^9 \]

Step 2: Simplify the Exponential Equation

Calculating $3^{-2}$: \[ 3^{-2} = \frac{1}{3^2} = \frac{1}{9} \] Thus, we have: \[ \frac{1}{9} = x^9 \]

Step 3: Solve for $x$

To find $x$, we take the ninth root of both sides: \[ x = \left(\frac{1}{9}\right)^{\frac{1}{9}} = 9^{-\frac{1}{9}} \] Calculating this gives us approximately: \[ x \approx 0.7834 \]

Final Answer

\[ \boxed{x \approx 0.7834} \]

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