query: Show that $e^{\ln(z)}$ always equals $z$, but $\ln(e^z)$ does not always equal $z$.
positive_passages.docid: 146873
positive_passages.text: Why is $\log(e^z)\ne z$ in complex analysis?
negative_passages.docid: 22303
negative_passages.text: How do I prove that $f(x)f(y)=f(x+y)$ implies that $f(x)=e^{cx}$, assuming f is continuous and not zero?
negative_passages.docid: 127538
negative_passages.text: Show $f(x)=x\ln x$ is not uniformly continuous
negative_passages.docid: 123467
negative_passages.text: Prove that $f(z) = e^z$ maps $\mathbb{C} \setminus B $ onto $\mathbb{C} \setminus \{0\}$.
negative_passages.docid: 92595
negative_passages.text: A problem regarding the in-equality of complex numbers .
negative_passages.docid: 125209
negative_passages.text: Prove $e^x - e^y \leq e |x-y|$ for $x$ belonging to $[0,1]$
negative_passages.docid: 14759
negative_passages.text: Prove that $ f(z) = 0 $ for all $ z \in \overline{D}(0,1) $.
negative_passages.docid: 104688
negative_passages.text: How can I prove that $x-{x^2\over2}<\ln(1+x)$
negative_passages.docid: 144539
negative_passages.text: Power series $ f(g(z)) = z $
negative_passages.docid: 141156
negative_passages.text: The domain of $\ln(x)^{\ln(x)}$
negative_passages.docid: 99821
negative_passages.text: Show that $u(x)=\ln\left(\ln\left(1+\frac{1}{|x|}\right)\right)$ is in $W^{1,n}(U)$, where $U=B(0,1)\subset\mathbb{R}^n$.
negative_passages.docid: 93408
negative_passages.text: Trying to show that $\ln(x) = \lim_{n\to\infty} n(x^{1/n} -1)$
negative_passages.docid: 53583
negative_passages.text: Why $e^x$ is always greater than $x^e$?
negative_passages.docid: 101701
negative_passages.text: Prove that $\ln x$ is not uniformly continuous on $(0,1]$
negative_passages.docid: 83713
negative_passages.text: If $z,w\in \mathbb{C}^{\times}$, then $|z+w|=|z|+|w|$ iff $w=tz$ for some $t\gt 0$
negative_passages.docid: 65054
negative_passages.text: How is $e^z$ the only function that is entire with the property $f'(z) = f(z)$ and $f(0) = 1$
negative_passages.docid: 102553
negative_passages.text: How to show that $f(z) = f(i)$