# Tagged Questions

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### On the equation $\exp(a x+b)=\ln(x)$

I am confronted with: $$\exp(a x+b)=\ln(x)$$ for $a,b$ reals and $a<0$, $b>0$. I need the (unique) solution for $x$. My first target is (if it exists) an analytic solution in terms of ...
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### Value of $(-1)^x$ for $x$ irrational

I was working on an analysis problem when this question arose in one my proofs. Obviously it's either $-1$ or $1$, but it seems like there can only be an arbitrary way to assign this. So is there an ...
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### Power function inequality

Let $x$ and $p$ be real numbers with $x \ge 1$ and $p \ge 2$ . Show that $(x - 1)(x + 1)^{p - 1} \ge x^p - 1$ . I recently discovered this result. I am sure it is known, but it is new to me. It is ...
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### Proof of exponential from homomorphism property

I am going back through a bunch of calculus I learned in high school and proving the stuff that they just told us was true. Along the way, I found I had to prove that if $f(x+y)=f(x)f(y)$ then $f$ is ...
### Simple estimation $e^{a\sqrt{r}} > r$
I want to prove a simple theorem about contour integration via residues and I need the following estimation: $e^{a\sqrt{r}} > r$ for any real a > 0 and r >> 0. Is this true? If so, what is an ...
### Zero to the zero power - Is $0^0=1$?
Could someone provide me with good explanation of why $0^0 = 1$? My train of thought: $x > 0$ $0^x = 0^{x-0} = 0^x/0^0$, so $0^0 = 0^x/0^x = ?$ Possible answers: $0^0 * 0^x = 1 * 0^x$, so ...