2^x=9^x-9

Take logs of both sides and use the fact that log(a^b) = b*log(a):

x*ln(2) = (x-9)*ln(9)

9*ln(9) = x*(ln(9) - ln(2))

x = 9*ln(9)/(ln(9) - ln(2))

$${\mathtt{x}} = {\frac{{\mathtt{9}}{\mathtt{\,\times\,}}{ln}{\left({\mathtt{9}}\right)}}{\left({ln}{\left({\mathtt{9}}\right)}{\mathtt{\,-\,}}{ln}{\left({\mathtt{2}}\right)}\right)}} \Rightarrow {\mathtt{x}} = {\mathtt{13.147\: \!608\: \!785\: \!565\: \!331\: \!8}}$$

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I think this is supposed to be

2^x=9^(x-9 )  

This is probably best solved by a graph........https://www.desmos.com/calculator/kpafbtwqix

The intersection point is about (13.15, 9122)     ....... your answer is probably better, since Desmos rounds to 2 decimal places........!!!