Student Peter solved the exponential equation $$ 10^x+100=10 $$ in the following way:
(1) He rewrote the numbers $100$ and $10$ as powers of $10$: $$ 10^x+10^2=10^1 $$
(2) He simplified the left side of the equation: $$ 10^{x+2}=10^1 $$
(3) From the equality of powers with the same base, he concluded: $$ x+2=1 $$
(4) He solved the resulting equation: $$ x=-1 $$
(5) He did the check of his solution and found that: $$ \begin{aligned} L & = 10^{-1}+100=100.1 \cr R & = 10 \cr L & \neq R \end{aligned} $$ Then he stated that the equation does not have any solution.
Did Peter make a mistake? If yes, identify the step where the mistake is.
Yes. The mistake is in step (1). The left side of the equation is in a form of a sum, so the number $100$ cannot be rewritten as $10^2$.
Yes. The mistake is in step (2). The equality $10^x+10^2=10^{x+2}$ does not generally hold.
Yes. The mistake is in step (3). The mentioned simplification can only be made when the base is less than $1$.
Yes. The mistake is in step (4). The solution should have been $x=-\frac12$.
Yes. The mistake is in step (5). When doing a check, it should have been: $$ L =10^{−1}+10^2=10^1= 10 $$
No. The whole procedure is correct.
To solve this equation, knowledge of the rules for exponentiation is essential. When we add powers with the same base, it is not possible to sum their exponents. Such an operation can only be used when we multiply powers of the same base. The correct solution is as follows: $$ \begin{gather} 10^x+100=10 \cr 10^x=-90 \end{gather} $$ The exponential expression $10^x$ is not negative for any real number $x$, and so the equation has no solution. In this case, performing the check is not necessary.