To solve the equation \( 9^x = x^6 \), we can start by rewriting \( 9 \) as \( 3^2 \). Therefore, we can rewrite the equation as:
\[ (3^2)^x = x^6 \]
This simplifies to:
\[ 3^{2x} = x^6 \]
Next, we can take the logarithm of both sides. Using the natural logarithm (though you can use any logarithm), we get:
\[ \ln(3^{2x}) = \ln(x^6) \]
Using the properties of logarithms, we can simplify both sides:
\[ 2x \ln(3) = 6 \ln(x) \]
Now we can rearrange this equation:
\[ \frac{2 \ln(3)}{6} = \frac{\ln(x)}{x} \]
This simplifies to:
\[ \frac{\ln(3)}{3} = \frac{\ln(x)}{x} \]
To solve this, we can set \( k = \frac{\ln(3)}{3} \) and look for solutions to the equation:
\[ \ln(x) = kx \]
This type of equation can be solved graphically or numerically since it cannot be solved algebraically. The function \( f(x) = \ln(x) - kx \) can be analyzed for roots.
We can start finding numerical solutions with some trial values of \( x \):
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If \( x = 1 \): \[ \ln(1) = 0 \quad \text{and} \quad k \cdot 1 = \frac{\ln(3)}{3} \approx 0.366 \] Not a solution.
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If \( x = 3 \): \[ \ln(3) \approx 1.0986 \quad \text{and} \quad k \cdot 3 = \ln(3) \approx 1.0986 \] This gives us a solution since both sides are equal.
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Values around \( x = 1 \) and \( x = 3 \) can be tested and narrowed down further to identify if more solutions exist or confirm that we only have one solution, which can be done with numerical analysis or numerical solvers.
Therefore, the solution we find is:
\[ \boxed{3} \]
This can be verified by substituting \( x = 3 \) back into the original equation. You might also consider numerical methods if you prefer to check for additional roots or express the function graphically.