A teacher asked his students to carefully study the following procedure for solving the exponential equation: $$3^{x+1}-3^x=3^x+9$$
1) We modify the right-hand side of the equation: $$3^{x+1}-3^x=3^x+3^2$$
2) There are now powers with base $3$ on both sides of the equation. If we cancel out the bases, we get the following relation for exponents: $$x+1-x=x+2$$
3) The equality $x+1-x=x+2$ holds if and only if: $$x=-1$$
4) It is not necessary to do the check in this case.
Is there a mistake in any steps? If yes, identify where.
Yes. There is a mistake in step (1). It should have been $3^{x+1}-3^x=3^{x+2}$.
Yes. There is a mistake in step (2). It is not possible to cancel out the bases in this case.
Yes. There is a mistake in step (3). It should have been $x=\frac12$.
Yes. There is a mistake in step (4). Performing the check is an integral part of the solving procedure.
No. The whole procedure is correct.
The correct solution of the exponential equation: $$ 3^{x+1}-3^x=3^x+9 $$
1) We modify the expression $3^{x+1}$ on the left-hand side of the equation, applying the rules for exponents: $$ 3^x 3^1-3^x=3^x+9 $$
2) We subtract the expression $3^x$ from both sides of the equation: $$ 3^x 3^1-3^x-3^x=9 $$
3) We factor out on the left-hand side and solve the equation: $$ \begin{aligned} 3^x (3^1-1-1) & = 9 \cr 3^x & =9 \cr 3^x & =3^2 \cr x & =2 \end{aligned} $$