I'm trying to see why the below is true.  $$ x^{\log(a)} = a^{\log(x)} $$  Anyone here know why this is? Thank you.
I have been hit with a homework problem that I just have no idea how to approach. Any help from you all is very much appreciated. Here is the problem  Prove the equation: $a^{\log_b c} = c^{\log_b a}$  Any ideas?
Maximize window shortcut What is the keyboard shortcut to maximize a window on OSX? I'd like to know the shortcut that is equivalent to double clicking on the window title. I do not want to make the window full screen. Prove,by vector method,that the point of intersection of the diagonals of the trapezium lies on the line passing through the mid-points of the parallel sides.    My Attempt: Let the trapezium be $OABC$ and that the O is a origin and the position vectors of $A,B,C$ be $\vec{a},\vec{b},\vec{c}$.Then the equation of $OB$ diagonal is $\vec{r}=\vec{0}+\lambda \vec{b}................(1)$ And equation of $AC$ diagonal is $\vec{r}=\vec{a}+\mu(\vec{c}-\vec{a}).......(2)$ And the equation of the line joining the mid points of $OA$ i.e.$\frac{\vec{a}}{2}$ and $BC$ i.e. $\frac{\vec{b}+\vec{c}}{2}$ is $\vec{r}=\frac{\vec{a}}{2}+t(\frac{\vec{b}+\vec{c}}{2}-\frac{\vec{a}}{2}).........(3)$. Here $\lambda,\mu,t$ are scalars. I do not know how to solve $(1)$ and $(2)$ and put into $(3)$ to prove the desired result. Please help me.Thanks. How to show google.com in an iframe? Derivative of conjugate multivariate function What is wrong with "Where should this car be parked?"? Launching an Applet fails? Here is the proof I tried to weave while trying to prove this theorem:     Theorem. Every infinite subset of a countable set is countable.      Proof. Let $A$ be a countable set and $E\subset A$ be infinite. Then $A\thicksim\mathbb{N}$. This implies that there is a sequence $\left\{x_{n}\right\}_{n\in\mathbb{N}}$ where $x_{n}\in A$. Construct a sequence $\left\{n_{k}\right\}_{k\in\mathbb{N}}$ where $x_{n_{k}}\in A$, $n_{1}$ is the smallest positive integer such that $x_{n_{1}}\in A$, and $n_{k+1}&gt;n_{k}$. Then      $$ E=\bigcup_{k=1}^{\infty}x_{n_{k}}, $$      and $E$ is countable, because $E\thicksim\mathbb{N}$. $\blacksquare$   Is it convincing? On our minecraft server, I've built a Nether portal in my basement. Which connected two-way (tested) to a location in the Nether.    This nether portal took me to an area in the nether which was a small cave. I dug upwards and surfaced close to a Nether Fortress. After clearing the fortress of Blaze spawners, I decided to 'move in'. So I went back to the portal, dismantled it, and reactivated in in a location in the Fortress.    The fortress would be within the same 128m radius I would assume as it's really close to the original place. Literally some steps to the surface, then a ladder up the nearest Nether Fortress bridge support.  Now, taking this portal back to my original doesn't work, I spawn deep underground in an Abandoned Mine Shaft.  I read somewhere that my Nether portal has found a 'better match' and this could be remedied by disabling the 'wrong' portal in the mine shaft.  However, upon multiple attempts, a new nether portal is dynamically created in the same area. See screenie for example. You'll see two 'stacked' portals on the left (from my different attempts at disabling) and then the current active one on the right.    My question is, how can I make sure these portals go to the correct place. Is it a matter of returning the portal to it's original location or am I stuck for returning to the basement? If one is asking for a person, what should s/he say&mdash;1 or 2?   Could I speak to Kristina? Could I speak with Kristina? Why haven't I received the "Curious" badge? As you can see in the picture, I have 13 out of 5. But it's been like this for about two months! What's the reason for that? Why did my windshield freeze instantaneously? It was very cold outside, this morning, when I took the car that slept in the snow, with a simple cloth on the windshield. I entered the vehicle, drove a kilometer or so. The air inside was so cold I could see my breath (or maybe I forgot to brush my teeth). Warm air was blowing on the windshield from the inside and suddenly…  The glass in front of my became opaque, starting from the bottom (the place where the hot air was blowing) and freezing up, up, up until the whole screen was filled in about 5 seconds. It reminded my a little bit of how some baterias spread.  When I tried to remove the mist, I realised it was ice in the car. My windshield instant froze.  I know the theory behind supercooling: a very cold and still liquid can freeze when moved. I'm not sure what happened here, but there were some pretty big turns in the road before it froze. It really looked like the hot air froze the place and not the movement.  Any idea about what happened ? Is there a documentclass that produces 'endless' pages? Does anybody know of a documentclass that produces a page that has a fixed width and a variable length, depending on the actual content that needs to fit on the sheet? I'm trying to get something like a kind of scroll if possible. I normally use the following idiom to check if a String can be converted to an integer.  public boolean isInteger( String input ) {     try {         Integer.parseInt( input );         return true;     }     catch( Exception e ) {         return false;     } }   Is it just me, or does this seem a bit hackish?  What's a better way?    See my answer (with benchmarks, based on the  by ) to see why I've reversed my position and accepted  to this problem.  I think this original code will be used by most people because it's quicker to implement, and more maintainable, but it's orders of magnitude slower when non-integer data is provided. How to derive this binomial identity? Google chrome is reading characters backwards Google Chrome is my favorite browser, and I can't imagine going on without it. Suddenly (without closing and re-opening) the browser; chrome has been reading characters upside down, very weird:    Chrome version:    I tried restarting my browser and my MAC (running OS X Yosemite 10.10), but the error keeps on coming. How do I turn on SharePoint App store in SharePoint 2013 I am getting this message when I want to add a app:     Sorry, apps are turned off. If you know who runs the server, tell them   to enable apps. Is there any Stack Exchange site where I can ask others to recommend me good web sites for solving specific problems?  For example I would like to know if there is any URL shortening website that can allow me to define the url.  For example instead of http://bit.ly/1gX2dD0 I would like it to be http://bit.ly/superflow - if "superflow" is not available then I will invent another word.  I can imagine that such services are paid but I don't need the link to last more than one day, and maybe there is such a website. What are the use cases of SO_REUSEADDR? I have used SO_REUSEADDR to have my server which got terminated to restart with out complaining that the socket is already is in use. I was wondering if there are other uses of SO_REUSEADDR? Have anyone used the socket option for other than the said purpose? Why is learning modern algebraic geometry so complicated? Why is scale factor inversely proprotional to temperature? In all lectures, books and papers I have read about, the scale factor of the universe is inversely proportional to temperature  $$a \propto \frac{1}{T}$$  What is the reasoning behind this relation?  People did mention stretching of wavelength proportional to scale factor and wavelength being inversely proportional to temperature by Wein's law, but somehow this argument doesn't seem convincing to me. How did we initially come about to this relation?