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?
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