What happened to the solutions of this logarithmic equation?

Our math class is talking about logarithms now, and we came across a really interesting problem while solving a logarithmic equation: 2^^(log2(x^^2))=4, log2 means log of base 2, m^^n means m to the nth power.

Here’s how I solved it:

1. 2^^(log2(x^^2)) = 4

2. log2(4) = log2(x^^2)

3. 4 = x^^2

4. x = +/- 2

Here’s how the other person solved it:

1. 2^^(log2(x^^2)) = 4

2. 2^^(log2(x^^2)) = 2^^2

3. log2(x^^2)= 2

4. 2 log2(x)=2

5. log2(x) = 1

6. x=2

One of the solutions vanished! We thought it had something to with the property used in line 3 of her proof: log(X^^c) = c log(X). log(X^^c) = 0 has c solutions by the Fundamental Theorem of Algebra, while c log(x) = 0 only requires one. This property seems to deplete the number of solutions. How can one justify this phenomenon? It doesn’t seem mathematically sound to me. Where did the solutions go?