Carefully study the following steps taken in solving the exponential equation: $$ 6^{1+x}+6^{1+x}=12 $$
1) First, combine the terms on the left side of the equation: $$ 2 \cdot 6^{1+x}=12 $$
2) Then, adjust the left and right sides of the equation so that both sides have a power with the same base of $12$: $$ 12^{1+x}=12^1 $$
3) Since the bases on both sides are equal, the exponents must also be equal: $$ 1+x=1 $$
4) The equality holds for: $$ x=0 $$
It is not necessary to perform a check. Nevertheless substituting back confirms correctness of the solution.
Is there a mistake in any of these steps? If so, identify the step.
Yes. There is a mistake in step (1). The equality $6^{1+x}+6^{1+x}=2\cdot 6^{1+x}$ does not hold in general. It should be: $$6^{1+x}+6^{1+x}=6^{2+2x}$$
Yes. There is a mistake in step (2). The adjustment of the left side $2\cdot 6^{1+x}=12^{1+x}$ of the equation is incorrect.
Yes. There is a mistake in step (3). The equality $12^{1+x}=12^1$ implies $1+x=1$ or $1+x=0$. Because the case $1+x=0$ was not considered not all the solutions were found.
No. There is not a mistake. The entire procedure is correct.
The correct procedure of solving the exponential equation: $$6^{1+x}+6^{1+x}=12$$
1) Sum the terms on the left side of the equation: $$ 2\cdot 6^{1+x}=12 $$
2) Divide the entire equation by $2$ and adjust both sides of the equation to have powers with the same base: $$ 6^{1+x}=6^1 $$
3) Since the bases are equal, the exponents must also be equal: $$ 1+x=1 $$
4) The equality holds if and only if: $$x=0$$
Note: Since only equivalent transformations were used during the solving procedure, a follow-up check is not necessary.