explain the following paradox that bothered mathematicians o

explain the following paradox that bothered mathematicians of Euler\'s time: since (-x)^2 = (x)^2, we have log(-x)^2 = log(x)^2, whence 2log(-x) = 2log(x), and thence log(-x) = log(x)

Solution

The domain of the log function is positive real numbers.

Therefore, if x > 0, then -x < 0, thus log(-x) is undefined. If -x > 0, the x < 0, thus log(x) is undefined.