This is a basic notation question. With trig functions like $\sin{x}$ and $\cos{x}$, if they are raised to a power, let's say $a$, then we write it, in the case of $\sin{x}$, as $\sin^a{x}$. However, with regards to logarithms, I've been wondering if the notation is the same; is $(\ln{x})^a$ written as $\ln^a{x}$? That'd simplify and clarify some working significantly. Thank you for your help in this trivial matter.
What is it used for and why  doesn't it equal $\log{x^2}$?
Whenever adding a new wireless network, android seems to forget the network after disabling then re-enabling wifi. I am forced to manually connect to the network and enter the password each time. This also happens with open networks. The problem occurs on any wireless network, at home, work, public networks, etc.  Further inspection on the /data/misc/wifi/wpa_supplicant.conf file shows "disabled=1" added to the end of each network.    click image for larger version  This has been going on for weeks. Any help would be much appreciated.  Phone: Nexus 4 Version: 4.4.4 Read echo'ed output from another PHP file How to know current name of the database in Django? Why does nothing show up in the Compositing tab? I am trying to follow Blender Guru's beginner tutorial. He went to the Compositor tab and was using it to Denoise an image but when I got there, no nodes show up. Can you guys help? Why isn't my website in Google search results? It's been sometime and Google has yet to index my site. How can I get Google to index my website? So, to explain the title, I'm referring to the necessity of the axiom of choice in the existence of a well ordering on reals, or any uncountable set.  Now, while tweaking some sets, I came across this :  We start with the natural numbers, $N$. We take the power set of the naturals, $P(N)$.  Then we remove all the finite subsets of $N$ from $P(N)$. Let us call this new set $S$. This set is the set of all infinite subsets of $N$.  It is easy to show that $S$ has uncountable cardinality, same as that of real numbers. This is because the removal of finite subsets only removes a countable number of elements. (Haven't posted this deduction, for it is very easy, but I may post it if it is not so evident)  Now, we seek to find an ordering on the set $S$. Every set in this set is an infinite subset of natural numbers, so each of these sets are well ordered by the natural ordering of $N$.  Taking any two sets in $S$, say $A$, and $B$, we seek to order them by checking their elements lexicographically. We compare the first two elements in $A$, and $B$. Let them be $a_1$, and $b_1$ respectively. If $a_1 = b_1$, then we move on to the second elements in the sets, $a_2$, and $b_2$, and so on. If, at any point, $a_n &lt; b_n$, then $A &lt; B$, or if $b_n &lt; a_n$, then $B &lt; A$.  This order seems to be a well ordering of the uncountable infinity of reals. I don't seem to have invoked the axiom of choice anywhere in the construction of this set $S$.  So, why isn't this a well ordering on the uncountable of reals? Should "move-to-chat" be provided on demand (as a button perhaps)? How to disable X server autostart in Debian Jessie? Given that not all employees have a diamond:     All Stack Overflow employees aren't given a moderator diamond; the Community team has tightened up internal policies regarding who gets one and who doesn't. [...] —      can we have some visual indication that they are an employee?  This was  back in 2011, but declined. However, the situation appears to have changed now with fewer employees being given a diamond.  That way when they post on a meta site we know that it's somewhat "official".  Failing that please ensure that everyone includes the fact they're an employee in their profile given that  to add employee to their profile pages was also declined. Does the (relativistic) mass change? Why? I learned recently that when an object moves with a velocity comparable to the velocity of light the (relativistic) mass changes. How does this alteration take place? I am using pgfplot to draw graphs in latex. Now this is the problem I am facing:  \begin{tikzpicture}[scale=1] \begin{axis}[ybar,grid=major,bar width=2pt,width=8cm, height=8cm,xmin=1,xmax=540,ymin=1,ymax=240] \addplot [draw=black,fill=green]  table [x=size, y=cluster, col sep=comma] {sim-0.4-0.6.csv}; \end{axis} \end{tikzpicture}   It generates the following graph:    Since, I have values mainly between 0-50 on x-axis, I want to zoom there. After that, values comes around 500. S0, there are no graph points between 50 - 450. How should I do that? What is the best way to close an ASPX page from the code-behind?    I have a button event handler that I want to close the page after the user has clicked an ASP.NET button on the page.  I have tried to programmatically add a JavaScript method that contains a window.close() command to the OnClientClick event to close the page but it does not work.  The button is also a asp:AsyncPostBoskTrigger for an AJAX Update Panel.  The application uses .NET Framework 3.5. Which nvidia driver from additional drivers option should I choose? I have NVIDIA Geforece 7025 / nForce 630a integrated gfx card on my UBUNTU 12.04.And I have 4 options in Additional drivers:    NVIDIA accelerated graphics driver (version 173)   NVIDIA accelerated graphics driver (post-release updates)(version current updates)  NVIDIA accelerated graphics driver (version current)[Recommended]  NVIDIA accelerated graphics driver (post-release updates)(version 173 updates)   I am currently using the First option.  My Problem is there I can't update the version 173 to the latest version of drivers using xswat PPA. Whenever I do it and reboot but the NVIDIA xserver sttings always shows me the same version.  Which option would allow me to update my drivers to the current version? Drupal is fast at develop, but the performance is so poor.  It is hard to reach 50 request per second.  And there are so many SQL queries in a simple web page.  If you test the request time of a form submit in Drupal.org it will always take seconds to finish.  How do you improve the speed of your web site? How do I change which audio jacks are used for input and output? I'm using a Realtek HD audio card built-in my motherboard. The Windows driver comes with a control panel that allows me to select which back panel jacks are used for what. So for example I can make both the blue jack and green jack for output and only the red one for mic-in. (Whereas by default, the blue jack is for line in, which I never need.)  How can I do the same under Linux? If possible, please don't suggest something that involves PulseAudio or JACK; I'd like to do it the plain way, e.g. by editing ALSA configuration files, if possible. The way I understand it, my problem should have nothing to do with software servers redirecting streams, just instructing the driver to treat this jack as so and so because it's hardware supported.  Thank you very much! The idea behind the sum of powers of 2 I know that the sum of power of $2$ is $2^{n+1}-1$, and I know the mathematical induction proof. But does anyone know how $2^{n+1}-1$ comes up in the first place.  For example, sum of n numbers is $\frac{n(n+1)}{2}$. The idea is that we replicate the set and put it in a rectangle, hence we can do the trick. What is the logic behind the sum of power of $2$ formula? Is there an IDictionary implementation that, on missing key, returns the default value instead of throwing? The indexer into Dictionary throws an exception if the key is missing.  Is there an implementation of IDictionary that instead will return default(T)? I know about the TryGetValue() method, but that's impossible to use with LINQ. Would this efficiently do what I need?: myDict.FirstOrDefault(a =&gt; a.Key == someKeyKalue);  I don't think it will as I think it will iterate the keys instead of using a Hash lookup. Show that $\textbf{ZF}$ shows $\phi$ is $\Delta_1$ In a multiple linear regression, why is it possible to have a highly significant F statistic (p&lt;.001) but have very high p-values on all the regressor's t tests?  In my model, there are 10 regressors. One has a p-value of 0.1 and the rest are above 0.9    For dealing with this problem see the . What is the difference between "why are you worried" and "why you are worried"?