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

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Question

(B) The domain of the natural logarithm function $\log x$ is the set of all positive real numbers,i.e.,$x \in (0, \infty)$.For $x \leq 0$,the expressi

Vedclass · Accepted Answer

No,it is true only for $x > 0$.

Vedclass · Answer

(B) The domain of the natural logarithm function $\log x$ is the set of all positive real numbers,i.e.,$x \in (0, \infty)$.For $x \leq 0$,the expression $\log x$ is not defined in the set of real numbers.For $x > 0$,let $y = e^{\log x}$.Taking the natural logarithm on both sides,we get $\log y = \log(e^{\log x})$.Using the property $\log(a^b) = b \log a$,we have $\log y = (\log x) \cdot \log e$.Since $\log e = 1$,we get $\log y = \log x$.By the one-to-one property of the logarithmic function,$y = x$.Therefore,the equation $x = e^{\log x}$ is true only for $x > 0$.

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

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