$$ 3 x^{x}-3^{x+10} $$; Find $$ x $$

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To solve the equation: $$ 3x^{x} - 3^{x + 10} = 0 $$ we need to find the value of $$ x $$ that satisfies this equation. ### Step 1: Simplify the Equation First, let's rewrite the equation: $$ 3x^{x} = 3^{x + 10} $$ Next, divide both sides by 3 to simplify: $$ x^{x} = 3^{x + 9} $$ ### Step 2: Take the Natural Logarithm Taking the natural logarithm of both sides helps to linearize the equation: $$ \ln(x^{x}) = \ln(3^{x + 9}) $$ Simplify using logarithm properties: $$ x \ln(x) = (x + 9) \ln(3) $$ Rearrange terms: $$ x \ln(x) - x \ln(3) = 9 \ln(3) $$ $$ x (\ln(x) - \ln(3)) = 9 \ln(3) $$ ### Step 3: Analyze the Equation Let's consider the function: $$ f(x) = x (\ln(x) - \ln(3)) $$ We need to find $$ x $$ such that: $$ f(x) = 9 \ln(3) $$ Testing $$ x = 9 $$: $$ f(9) = 9 (\ln(9) - \ln(3)) = 9 (\ln(3^2) - \ln(3)) = 9 (2\ln(3) - \ln(3)) = 9 \ln(3) $$ This matches the right-hand side of the equation. Therefore, $$ x = 9 $$ is a solution. ### Step 4: Uniqueness of the Solution To ensure that $$ x = 9 $$ is the only solution: - For $$ x < 3 $$, $$ \ln(x) < \ln(3) $$, making $$ f(x) $$ negative. - For $$ x = 3 $$, $$ f(x) = 0 $$. - For $$ x > 3 $$, $$ f(x) $$ increases monotonically because both $$ \ln(x) $$ and $$ x $$ increase. Since $$ f(x) $$ is increasing for $$ x > 3 $$ and $$ f(9) = 9 \ln(3) $$, there are no other solutions beyond $$ x = 9 $$. ### Conclusion The value of $$ x $$ that satisfies the equation is: $$ \boxed{9} $$ $$ x = 9 $$

\( 3 x^{x}-3^{x+10} \); Find \( x \)

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