Hi Malavika,
The given equation is,
$$9^{2x-1}\ -\ 81\ ^{x-1}\ =\ 1944$$
by substituting $$9^2$$    in place of 81, we get
$$9^{2x-2+1}\ -\ 9\ ^{2\left(x-1\right)}\ =\ 1944$$
by taking  $$9^{2x-2}$$ common we get,
$$9^{2x-2}\left(9\ -\ 1\right)\ =\ 1944$$
$$9^{2x-2}\ =\ \frac{1944}{8}\ =\ 243$$
by substituting $$3^2$$ in place of 9, we get
$$3^{2\left(2x-2\right)}\ =\ 243\ =\ 3^5$$
which means that,
2(2x - 2) = 4x - 4 = 5
and x = $$\frac{9}{4}$$.
Hope this clears your doubt!

Hi,

We have updated the solution. Please check. Thanks for pointing it out.

Hi Malavika,
We are assuming the value of $$9^{2x}$$ to be a and we are not taking it as $$81^{\left(x-1\right)}$$.
If you look at the solution you will see that sir has taken,
 $$81^{\left(x-1\right)}$$ is taken as $$9^{\left(2x-2\right)}\ =\ \frac{9^{2x}}{81}\ =\frac{a}{81}$$
Hope this clears your doubt!