Is it true that x = elog x for all real x? - Vidyadip

Question

Is it true that x = e^{log x} for all real x?

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Answer

First, observe that the domain of log function is a set of all positive real numbers. So the above equation is not true for non-positive real numbers.
Now, let y = e^{log x}.
If y > 0, we may take logarithm which gives us
log y = log (e^{log x}) = log x . log e = log x.
Thus y = x.
Hence x = e^{log x} is true only for positive values of x.

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