Robert was tasked with solving a simple exponential equation: $$ \frac13(108−3^x)=3^x $$
He solved the equation in the following steps:
1) He started by simplifying the left side of the equation: $$ 36−3^x=3^x $$
2) Then, he added $3^x$ to both sides of the equation and combined the terms on the right side: $$ \begin{aligned} 36 & =3^x+3^x \cr 36 & =2 \cdot 3^x \end{aligned} $$
3) Finally, he expressed the terms on both sides of the equation as powers of the same base and compared the exponents to obtain the solution: $$ \begin{aligned} 36 & =6^x \cr 6^2 & =6^x \cr x & =2 \end{aligned} $$ Did Robert make any mistakes? If yes, identify where.
Yes. He made the mistakes in steps (1) and (3).
Yes. He made the mistakes in steps (1) and (2).
Yes. He made the mistakes in steps (2) and (3).
Yes. He made the mistake only in step (1).
Yes. He made the mistake only in step (3).
No. All steps are correct.
Let us see the correct solution of the equation: $$ \frac13(108-3^x)=3^x $$ After multiplying both side by $3$, we obtain: $$ 108-3^x=3 \cdot 3^x $$ Then, we can add $3^x$ to both sides of the equation and combine like terms on the right side: $$ \begin{aligned} 108 & =3 \cdot 3^x+3^x \cr 108 & =4 \cdot 3^x \end{aligned} $$ Next, dividing both sides by $4$, we get: $$ 3^x=27 $$ Finally, expressing $27$ as $3^3$ we have the exponential equation with the same base on both sides, and the solution is obtained by comparing the exponents. $$ \begin{aligned} 3^x & =3^3 \cr x & =3 \end{aligned} $$ Note: The student made the first mistake in step (1) by incorrectly expanding the parenthesis. He multiplied by $\frac13$ only the first term in the parenthesis. The second mistake he made was in step (3). The equality $2\cdot 3^x=6^x$ does not generally hold.