According to wikipedia, we have that $$ x^{\log_b(y)} = y^{\log_b(x)} $$ because $$ x^{\log_b(y)} = b^{\log_b(x) \log_b(y)} = b^{\log_b(y) \log_b(x)} = y^{\log_b(x)} $$ But what justifies that first leap? $$ x^{\log_b(y)} = b^{\log_b(x) \log_b(y)} $$

Good evening, I was wondering if there is such a valid operation as raising a matrix to the power of a matrix, e.g. vaguely, if $M$ is a matrix, is $$ M^M $$ valid, or is there at least something similar? Would it be the components of the matrix raised to each component of the matrix it's raise...

Consider the following theorem: "Every non-empty set of positive integers has a minimum element". The proof I usually see is one that uses contradiction, and does not seem like the easiest possible proof. I think there is an easier proof, and I wonder why I never see it. Does it contain an inva...

$L=\lim_{n\to \infty}\frac{1}{\sqrt[n]{n!}}$. Then (1)$L=0$ (2)$L=1$ (3)$0<L<1$ (4) $L=\infty$ Let $x_n=\frac{1}{\sqrt[n]{n!}}$. Taking logarithm on both sides, $\log(x_n)=\frac{1}{n}\sum_{k=1} log(\frac{1}{k})$ Using the Cauchy's first theorem on limits,$\lim_{n\to \...

When I see the answer to the problem. How to check the convergence of the series whose elements are taken from the set $A$? One question raised in my mind, Can I prove $\sum_{n=1}^{\infty}\frac{1}{n}$ diverges using the following technique? By the Prime decomposition theorem, $n=2^{j_1}3^{j_2}...

Let $R$ be a PID. An ideal $P$ in $R$ is said to be primary if $ab \in P$ and $a \notin P$ implies $b^n \in P$ for some $n \in \Bbb{N}$. Show that $P$ is primary if and only if $P = (p^n)$ for some $n \in \Bbb{N}$ and some prime element $p \in P$. Here is my attempt: Assume that $P = (p...