9 ^ x = x

[low parcel]

solve for x

[formal needle]

This equation has no real solution

Define $f(x) := 9^x - x$, then $f'(x) = \ln(9) \cdot 9^x - 1$.

[burnt sparrow]

ℝafain

[formal needle]

Hence
$$f'(x)
\begin{cases}
<0&\text{ if }x<-\frac{\ln(\ln(9))}{\ln(9)}\
=0&\text{ if }x=-\frac{\ln(\ln(9))}{\ln(9)}\

0&\text{ if }x=-\frac{\ln(\ln(9))}{\ln(9)}
\end{cases}$$

[burnt sparrow]

ℝafain

[formal needle]

Hence $f$ is minimum at $-\frac{\ln(\ln(9))}{\ln(9)}$, where its value is $\frac{1+\ln(\ln(9))}{\ln(9)} > 0$.

[burnt sparrow]

ℝafain

[formal needle]

Hence $f$ does not attain zero anywhere in $\mathbb R$.

[burnt sparrow]

ℝafain

[normal remnant]

the solution in C can be defined as follows:
-(W(-2ln(3)))/(2ln(3))

I love W Lambert tbh