<s>[INST] Solve fo a step by step $$\frac{81^{a}+9^{a}+1}{9^{a}+3^{a}+1}=\frac{7}{9} \Rightarrow a = ? $$
 [/INST] Hint: Let $x=3^a>0$. $$\frac{81^a+9^a+1}{9^a+3^a+1}=\frac{x^4+x^2+1}{x^2+x+1}=\frac{\left(x^2+1\right)^2-x^2}{x^2+x+1}$$
$$=\frac{\left(x^2+1+x\right)\left(x^2+1-x\right)}{x^2+x+1}=x^2+1-x=\frac{7}{9}$$
So you have a quadratic equation for $x$, and then $a=\log_3(x)$.
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