As part of his final exam in Mathematics, Peter was tested of solving three simple exponential and logarithmic equations:
Task 1: $7^{2x}=7^x$
Task 2: $\log_3 x=0 $
Task 3: $7^x=0$
Take a careful look at his solutions.
1) To solve the equation: $$ 7^{2x}=7^x $$
First, he cancelled the bases: $$ 2x=x $$
Next, he divided the resulting equation by $x$ and got: $$ 2=1 $$
Thus, he stated that the given equation has no solution.
2) To solve the equation: $$ \log_3 x=0 $$
He applied the rule $\log_z x=y \Leftrightarrow x=z^y$ and obtained: $$ \begin{aligned} x&=3^0 \cr x&=1 \end{aligned} $$
3) Petr looked at the third equation: $$ 7^x=0 $$ He claimed that it had no solution and justified this by stating that the expression $7^x$ is positive for all real $x$.
Did Peter make any mistakes in his solutions? If yes, identify where.
Yes. He made a mistake only in task 1.
Yes. He made a mistake only in task 2.
Yes. He made a mistake only in task 3.
Yes. He has made a mistake in more than one task.
No. He did not make a mistake.
Peter made the mistake in task 1. The resulting equation $2x=x$ is correct. However, it cannot be divided by $x$ because we do not know if $x$ is non-zero. The equation $2x=x$ should be solved as follows. By subtracting $x$ from both sides of the equation we obtain: $$2x -x=x-x$$ And then $$x=0$$
which is the (only) solution to the given equation.