$\log^2{⁡x}-2 \log{⁡x}=0$

Three students, Jana, Dana, and Alena, were tasked with solving the following equation for $x \in \mathbb{R}$. $$\log^2{⁡x}-2 \log{⁡x}=0$$

Examine the steps each of them took to solve the equation and determine which of them proceeded correctly.

Jana

(1) Determined the domain of the logarithm $\log⁡{x}$: $$ x>0 $$

(2) Divided the equation $\log^2{⁡x}-2 \log{⁡x}=0$ by $\log{⁡x}$: $$ \log{⁡x}-2=0 $$

(3) Moved the $2$ to the right side of the equation: $$ \log{x} = 2 $$

(4) Simplified the equation using the logarithmic identity: $$ \log_a{⁡x}=v \Leftrightarrow x=a^v $$ and obtained the solution: $$ x=10^2 $$

Alena

(1) Determined the domain of the logarithm $\log{⁡x}$: $$ x>0 $$

(2) Modified the equation $\log^2{⁡x}-2 \log{⁡x}=0$ using properties of logarithms to: $$ 2 \log{⁡x}-2 \log⁡{x}=0 $$ resulting in: $$0=0$$

(3) Concluded from the equation and the domain of the logarithm that the solution is the set of all positive real numbers.

Dana

(1) Determined the domain of the logarithm $\log{⁡x}$: $$x>0$$

(2) Into the equation $\log^2⁡{x}-2 \log{⁡x}=0$, introduced the substitution: $$\log{⁡x}=t$$

(3) Obtained the quadratic equation: $$t^2-2t=0$$

(4) Solved the quadratic equation by factoring $t$: $$ \begin{gather} t^2-2t=0 \cr t(t-2)=0 \end{gather} $$ yielding the roots: $$ t=0, ~t=2 $$

(5) Then, returned to the substitution and solved for the $x$ roots: $$ \begin{gather} t=0 \Rightarrow \log{⁡x}=0 \Rightarrow x=1 \cr t=2 \Rightarrow \log{⁡x}=2 \Rightarrow x=100 \end{gather} $$