To solve the equation \( x^x = 3^{x+9} \), we can start by rewriting both sides in terms of logarithms for better manipulation.
Taking the natural logarithm of both sides gives us:
\[ \ln(x^x) = \ln(3^{x+9}). \]
Using the properties of logarithms, we can simplify each side:
\[ x \ln(x) = (x + 9) \ln(3). \]
Now we have:
\[ x \ln(x) = x \ln(3) + 9 \ln(3). \]
Rearranging the equation yields:
\[ x \ln(x) - x \ln(3) = 9 \ln(3), \]
or
\[ x (\ln(x) - \ln(3)) = 9 \ln(3). \]
This implies:
\[ x = \frac{9 \ln(3)}{\ln(x) - \ln(3)}. \]
This equation is challenging to solve analytically, so we can explore numerical methods or specific values for \( x \) to find potential solutions.
Let's test some integer values for \( x \):
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For \( x = 3 \): \[ 3^3 = 27, \] \[ 3^{3+9} = 3^{12} = 531441. \] Not a solution.
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For \( x = 2 \): \[ 2^2 = 4, \] \[ 3^{2+9} = 3^{11} = 177147. \] Not a solution.
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For \( x = 4 \): \[ 4^4 = 256, \] \[ 3^{4 + 9} = 3^{13} = 1594323. \] Not a solution.
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For \( x = 1 \): \[ 1^1 = 1, \] \[ 3^{1 + 9} = 3^{10} = 59049. \] Not a solution.
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For \( x = 0 \): \[ 0^0 \text{ is generally considered to be } 1 \text{ in combinatorial settings, but it is undefined.} \]
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For \( x = 9 \): \[ 9^9 = 387420489, \] \[ 3^{9 + 9} = 3^{18} = 387420489. \] This works!
From the above exploration, we find that:
\[ x = 9 \]
is a valid solution to the equation \( x^x = 3^{x+9} \). However, this equation may have additional solutions; thus you could use numerical methods or graphing tools to fully explore the behavior of both sides of the equation for any other potential solutions.
The solution is:
\[ \boxed{9}. \]