Solve $\ln {x^2} = 2\ln 4$

Question

Solve the equation $\ln {x^2} = 2\ln 4$

What follows is my approach to the problem. I would appreciate any pointers on what is wrong with it.

\begin{align} \ln x^2 & = 2 \ln 4 \\ 2 \ln x & = 2 \ln 4 &&\text{$\log_a b^c = c\log_a b$} \\ \ln x & = \ln 4 \\ x & = 4 \end{align}

Answer 1

Gotcha! \begin{align} \ln x^2 & = 2 \ln 4 \\ \ln |x|^2 & = 2 \ln 4 \\ 2 \ln |x| & = 2 \ln 4 \end{align}

Answer 2

Your solution is correct, but not complete.

Following your approach: $$\ln x^2=\ln 4^2$$ $$x^2=16,$$

which has two solutions.

Answer 3

You missed the solution $x=-4$. Here are the steps: $$\ln x^2=2\ln 4 = \ln 4^2 \Longrightarrow x^2 = 4^2 \Longrightarrow x = \pm 4 $$