A paradox bothered mathematicians for years?

Test Preparation
1

<p>Hey, just want you guys to relax from the brutal SAT.
Can anyone explain why?</p>

<pre><code> (-x)^2 = (x)^2
log(-x)^2 = log(x)^2
2log(-x) = 2log(x)

              ?
  log(-x)   = log(x)

</code></pre>

2

<p>log is not defined for negative values. therefore either log(-x) or log(x) doesn't exist</p>

3

<p>But the whole process seems reasonable.
Can you tell which step is wrong?</p>

4

<p>log(-x) can't exist if x>0
log(x) can't exist if x<0
so either way it is not right
2log(-x)= 2log(x)
is wrong</p>

5

<p>x=y doesn't mean that log(x)=log(y). This has to do with the fact that e^x is injective in the real plane but not in the complex plane, and the definition of log(x).</p>

<p>The same logic applies to:
(-x)^2 = x^2 [take square roots]
-x = x</p>