All in the title basically. Not sure how to prove $a^{\log_a(b)}=b$.  Don't know how to use the rules to get that.
I'm currently doing a couple of exercises on logarithmic expressions, and I was a bit confused when presented with the following: $5^{\log_5 17}$.  The answer is $17$, which is the argument of the logarithm in the exponent, but I don't understand the reason why.  A previous question in the exercise was $12^{\log_{12} 144}$, which is pretty straight forward since $\log_{12} 144=2$ and $12^2=144$ but since $\log_5 17$ is an irrational number, I couldn't calculate it the same way.  So my question is, why is $x^{\log_x n}=n$?  [Update]  I've now managed to reason this out and realized how simple it is.  Basically, if $y=\log_x n$, then $y$ is the number that  $x$ must be raised to to become $n$; therefore, if I raise $x$ to that power $y$, I will naturally get $n$.  I guess my problem was that I tried to work out the question mechanically instead of intelligently and that's why I failed, because once I realized that I can't calculate $\log_5 17$, I was stumped.  Now that I reasoned it out, as explained in the previous paragraph, I understand it.
Is there a deb package to install Firefox ESR? I would like to have  on my system but I would like to install it via apt-get. Is there a special package to install this?  Download link would be: Why does the following behave unexpectedly in Python?  &gt;&gt;&gt; a = 256 &gt;&gt;&gt; b = 256 &gt;&gt;&gt; a is b True           # This is an expected result &gt;&gt;&gt; a = 257 &gt;&gt;&gt; b = 257 &gt;&gt;&gt; a is b False          # What happened here? Why is this False? &gt;&gt;&gt; 257 is 257 True           # Yet the literal numbers compare properly   I am using Python 2.5.2. Trying some different versions of Python, it appears that Python 2.3.3 shows the above behaviour between 99 and 100.  Based on the above, I can hypothesize that Python is internally implemented such that "small" integers are stored in a different way than larger integers and the is operator can tell the difference. Why the leaky abstraction? What is a better way of comparing two arbitrary objects to see whether they are the same when I don't know in advance whether they are numbers or not? Is Mana Weaving ok if it's followed by a thorough shuffle? My friends and I are relatively new to MtG. At some point, one of them introduced us to the idea of mana weaving, and we'd do this to declump our decks prior to a very thorough shuffling. However, at the Theros pre-release, one of us was asked not to do this, and we learned it was frowned upon and ceased to do it.  I just now came across , and it sounds like mana weaving's only bad if you use it to leave your deck in an even distribution, and it's OK if your deck is then thoroughly shuffled. In hindsight, the judge may have simply thought my friend wouldn't have shuffled afterwards, but at the time I came under the impression it was frowned upon even if you did shuffle afterwards.  So: is mana weaving my deck OK, if I shuffle it very thoroughly afterwards? Integrate $\int_{0}^{\pi}{\frac{x\cos{x}}{1+\sin^{2}{x}}dx}$ I installed the  on my Ubuntu&nbsp;11.04 (Natty Narwhal) with   sudo apt-get install lamp-server^   Now I navigate to /var/www and use   wget http://ftp.drupal.org/files/projects/drupal-7.x-dev.tar.gz   I get the following error.   --2011-08-03 13:59:00--   http://ftp.drupal.org/files/projects/drupal-7.x-dev.tar.gz Resolving  ftp.drupal.org... 64.50.233.100, 64.50.236.52 Connecting to  ftp.drupal.org|64.50.233.100|:80... connected. HTTP request sent,  awaiting response... 200 OK Length: 2757101 (2.6M)  [application/x-gzip] drupal-7.x-dev.tar.gz: Permission denied   Cannot write to `drupal-7.x-dev.tar.gz' (Permission denied).   How do I resolve this error?  I tried to download the same file in the downloads folder, and it went on without an issue. So it seems to be some permission error, but I'm not sure about that or how to solve it. Incorrect bookmarks and page number in table of contents I have problem with table of contents and with bookmarks. In pdf file   table of contents shows wrong page number of Introduction and Bibliography, Bookmarks: introduction and bibliography don't bring me to the correct page.   Part of my code  \documentclass[12pt,a4paper]{book}  \usepackage[polish]{babel} \usepackage{polski} \usepackage[cp1250]{inputenc} \usepackage[pdftex,pagebackref=false,draft=false,pdfpagelabels=false,pdfstartview=FitH,pdfstartpage=1,bookmarks=true,pdfauthor={LiN},pdftitle={PD},pdfsubject={Titlei},pdfkeywords={words},unicode=true]{hyperref}  \usepackage{tocloft}  \begin{document} \tableofcontents \addcontentsline{toc}{chapter}{Introduction} \chapter*{Introduction} \chapter{Chapter1} \section{Section} \addcontentsline{toc}{chapter}{Bibliography} \begin{thebibliography}{9999999}% \bibitem{1}Author, \emph{Title of book}, Publisher. \end{thebibliography} \end{document}   How can I change that? How to copy files into /usr/local/? (permission denied) I'm still a little new to Ubuntu and was wondering how I can copy files, in this case Aptana Studio, to /usr/local/ without being denied.   Is there a way to do it in terminal? Or a way to allow myself permission?  I should add that I am using 10.10 Desktop edition, not server. Cross-thread operation not valid: Control accessed from a thread other than the thread it was created on I have a scenario. (Windows Forms, C#, .NET)   There is a main form which hosts some user control. The user control does some heavy data operation, such that if I directly call the UserControl_Load method the UI become nonresponsive for the duration for load method execution. To overcome this I load data on different thread (trying to change existing code as little as I can)   I used a background worker thread which will be loading the data and when done will notify the application that it has done its work.   Now came a real problem. All the UI (main form and its child usercontrols) was created on the primary main thread. In the LOAD method of the usercontrol I'm fetching data based on the values of some control (like textbox) on userControl.    The pseudocode would look like this:  CODE 1  UserContrl1_LoadDataMethod() {     if (textbox1.text == "MyName") // This gives exception     {         //Load data corresponding to "MyName".         //Populate a globale variable List&lt;string&gt; which will be binded to grid at some later stage.     } }   The Exception it gave was     Cross-thread operation not valid: Control accessed from a thread other than the thread it was created on.   To know more about this I did some googling and a suggestion came up like using the following code  CODE 2  UserContrl1_LoadDataMethod() {     if (InvokeRequired) // Line #1     {         this.Invoke(new MethodInvoker(UserContrl1_LoadDataMethod));         return;     }      if (textbox1.text == "MyName") // Now it wont give an exception     {     //Load data correspondin to "MyName"         //Populate a globale variable List&lt;string&gt; which will be binded to grid at some later stage     } }   BUT BUT BUT... it seems I'm back to square one. The Application again  become nonresponsive. It seems to be due to the execution of line #1 if condition. The loading task is again done by the parent thread and not the third that I spawned.  I don't know whether I perceived this right or wrong. I'm new to threading.  How do I resolve this and also what is the effect of execution of Line#1 if block?  The situation is this: I want to load data into a global variable based on the value of a control. I don't want to change the value of a control from the child thread. I'm not going to do it ever from a child thread.   So only accessing the value so that the corresponding data can be fetched from the database. Algorithm to detect overlapping periods Prevent VLC from starting a new instance every time a media file is opened I have designed a VF page to create a new record in which I am trying to display the value of a pick-list field. I have set a default value for this pick-list field. When trying to create the record using the salesforce native page, the value with which pick-list loads is the one I set as default but when creating the same record using VF page, the pick-list loads with "--None--" as default value.  Can anyone suggest If this is salesforce related bug or Am I doing anything wrong ? Lets say I have a function defined as $f(x) = \sqrt {x^2}$. Common knowledge of square roots tells you to simplify to $f(x) = x$ (we'll call that $g(x)$) which may be the same problem, but it isn't the same equation. For example, say I put $-1$ into them:  $\begin{align} f(x) &amp;= \sqrt {x^2} \\ f(-1) &amp;= \sqrt {(-1)^2} \\ f(-1) &amp;= \sqrt {1} \\ f(-1) &amp;= 1 \end{align}$  $\begin{align} g(x) &amp;= x \\ g(-1) &amp;= -1 \end{align}$  thereby, we conclude that $f(x)$ and $g(x)$ do not produce the same results even though they are mathematically the same. This is also shown when we try to graph the functions:  :    :    :    From this, we can see that given $f(x) = \sqrt {x^2}$, when simplified is not the same as $f(x) = x$. So, is there any way to prove that $y = \sqrt {x^2}$ is not the same as $y = x$ for negative values, but is infact the same as $y = \mid x \mid$? I want to find a set of integer solutions of Diophantine equation: $ax + by = c$, and apparently $\gcd(a,b)|c$. Then by what formula can I use to find $x$ and $y$ ?   I tried to play around with it: $x = (c - by)/a$, hence $a|(c - by)$.     $a$, $c$ and $b$ are known. So to obtain integer solution for $a$, then $c - by = ak$, and I lost from here, because $y = (c - ak)/b$. I kept repeating this routine and could not find a way to get rid of it? Any hint?  Thanks, Chan How to automatically fetch missing dependencies when installing software from .deb? Do you want to come with? Can I come with?  I seem to hear this construction more often in recent years, but it still grates on my ear. I know it's often said that one shouldn't , but I class that rule along with  (i.e. — pedantic tosh). Is it always informal/casual speech to end a sentence with &quot;with&quot;? Was it ever thus? Support for hardware components How can I find out if Ubuntu supports a particular hardware component, that is, if it works in Ubuntu?  I am going to put together a computer and I am choosing components.   How do I know if Ubuntu supports this or that motherboard, or this or that sound card, or this or that graphics card, etc? How do I know that there is no conflict in Ubuntu between this and that hardware component?   What resources are there to help with such situations?  I know that there are computers with Ubuntu pre-installed and that there are  for whole computers but this is not what I'm looking for. I'm looking for resources to find out support for particular hardware components and conflicts among particular hardware components. example of discontinuous function having direction derivative Is there a function (non piece-wise unlike below) which is discontinuous but has directional derivative at particular point? I have a manual that says the function has directional derivative at $(0,0)$ but is not continuous at $(0,0)$. $$f(x,y) = \begin{cases} \frac{xy^2}{x^2+y^4} &amp; \text{ if } x \neq 0\\  0 &amp; \text{ if } x= 0 \end{cases}$$  Can anyone give me few examples which is not defined piece wise as above? Is floating point math broken? Consider the following code:   0.1 + 0.2 == 0.3  -&gt;  false   0.1 + 0.2         -&gt;  0.30000000000000004   Why do these inaccuracies happen? Can anyone suggest a book on using complex numbers to solve geometry problems? Can anyone suggest me a book which includes everything which is needed for solving simple geometry problems through it? I only know about simple calculations on complex numbers. How to remove illegal characters from path and filenames? I need a robust and simple way to remove illegal path and file characters from a simple string. I've used the below code but it doesn't seem to do anything, what am I missing?  using System; using System.IO;  namespace ConsoleApplication1 {     class Program     {         static void Main(string[] args)         {             string illegal = "\"M&lt;&gt;\"\\a/ry/ h**ad:&gt;&gt; a\\/:*?\"&lt;&gt;| li*tt|le|| la\"mb.?";              illegal = illegal.Trim(Path.GetInvalidFileNameChars());             illegal = illegal.Trim(Path.GetInvalidPathChars());              Console.WriteLine(illegal);             Console.ReadLine();         }     } }