I am trying to solve the following question but without any result so far:     Let $p_1, p_2, \ldots, p_t$ be different primes. Prove that $\log (p_1), \log (p_2), \ldots, \log (p_t)$ are linearly independent over $\mathbb{Q}$ that is, if $x_1,x_2,\ldots, x_t$ are rational numbers with    $$ x_1 \log (p_1) + x_2 \log (p_2) + \cdots + x_t \log (p_t) = 0 $$   then $x_1 = x_2 = \cdots = x_t = 0$.    We are supposed to use the following result:  Let $x \in \mathbb{Q}$ with $x &gt; 0$. Then there is a unique sequence of integers $(n_2,n_3,n_5, \ldots)$, almost all equal to $0$, such that $$ x = \prod_{p \text{ prime}} p^{n_p}.  $$   Any tips are welcome for solving it, thanks.
Suppose we have a set of primes $p_1,\dots,p_t$. Prove that  $\log p_1,\dots,\log p_t$ is linear independent over $\mathbb{Q}$. Now, this implies $ \sum_{j=1}^{t}x_j\log(p_j)=0 \iff x_1=\dots=x_t=0$.  I think I have to use that fact that every $q\in\mathbb{Q}$ can be written as $\prod_{\mathcal{P}}$, where $n_p$ is a unique sequence ($n_2$,$n_3$,$\dots$) with domain $\mathbb{Z}$. Here, $\mathcal{P}$ denotes the set of all integers.  Now how can I use this to prove the linear independency?
I am working on a web application where I want the content to fill the height of the entire screen.  The page has a header, which contains a logo, and account information. This could be an arbitrary height. I want the content div to fill the rest of the page to the bottom.  I have a header div and a content div. At the moment I am using a table for the layout like so:  CSS and HTML    #page {     height: 100%; width: 100% }  #tdcontent {     height: 100%; }  #content {     overflow: auto; /* or overflow: hidden; */ } &lt;table id="page"&gt;     &lt;tr&gt;         &lt;td id="tdheader"&gt;             &lt;div id="header"&gt;...&lt;/div&gt;         &lt;/td&gt;     &lt;/tr&gt;     &lt;tr&gt;         &lt;td id="tdcontent"&gt;             &lt;div id="content"&gt;...&lt;/div&gt;         &lt;/td&gt;     &lt;/tr&gt; &lt;/table&gt;     The entire height of the page is filled, and no scrolling is required.  For anything inside the content div, setting top: 0; will put it right underneath the header. Sometimes the content will be a real table, with its height set to 100%. Putting header inside content will not allow this to work.  Is there a way to achieve the same effect without using the table?  Update:  Elements inside the content div will have heights set to percentages as well. So something at 100% inside the div will fill it to the bottom. As will two elements at 50%.  Update 2:  For instance, if the header takes up 20% of the screen's height, a table specified at 50% inside #content would take up 40% of the screen space. So far, wrapping the entire thing in a table is the only thing that works. How to pass arguments to addEventListener listener function? A better answer was posted long after accepting another. Should I accept the new one? Is the Macbook 12 inch's USB-C charger interchangeable with other USB-C chargers or devices? If there is only the Macbook 12 inch USB-C charger nearby, can it be used to charge other USB-C devices, such as any Android phones that also use USB-C? (will it be "too powerful" and damage other devices?)  Vice versa, if there is a Quick Charge 2.0 or 3.0 USB-C charger from other devices nearby, can it be used to charge the Macbook 12 inch?  The main concern is about whether it could damage the device or battery. I know in TNG that all computer interfaces are now touch-screen, and that got me thinking: when was the first reference to touch-screen computer interfaces in Sci-Fi made? How to install XAMPP? Hi all im a noob on Ubuntu/Linux World,  So i need some that give a full tutorial to How to install  from apachefriends.org correctly on Ubuntu 11.10 + all beyond Ubuntu for 32 bit and also 64 bit Desktop Version.. and also i need this:   Make a Folder on Your Home Folder and Linked to htdocs from Apache Web Server Make a GUI interface for Xmapp Start Xampp when Ubuntu Boot Up or Start Ubuntu   (Note: i ask it for both X86 and X64 so people in the further can see the tutorials)    New Data:  What i did was i download from  for now and i also what to say that I'm currently running Ubuntu 11.10 X64 bit for Dell Studio 1558 processor intel i3  i will let you know guys which solution was the best, and it is for a Desktop Version of Ubuntu 11.10 (Laptop 15 inch screen ^_^ ) not Server Ubuntu Switches needed in an n-location multiway switching system we can construct a two location multiway switching system with two 3-way switches:    They also teach us how to perform three location multiway switching with two 3-way switches and one 4-way switch:    Is it possible to make a similar system (three locations, all able to switch the light in any situation) with only three 3-way switches?   What can be said in general about the switches needed to make an n-location multiway switching system? I was about to add an extra signal handler to an app we have here and I noticed that the author had used sigaction() to set up the other signal handlers. I was going to use signal(). To follow convention I should use sigaction() but if I was writing from scratch, which should I choose? Suppose $ \alpha, \beta>0 $. Compute: $ \int_{0}^{\infty}\frac{\cos (\alpha x)-\cos (\beta x)}{x}dx $ Suppose $ \alpha, \beta&gt;0 $. Compute:   $$ \int_{0}^{\infty}\frac{\cos (\alpha x)-\cos (\beta x)}{x}dx $$   Here is what I do: $$\begin{align} \int_{0}^{\infty}\frac{\cos (\alpha x)-\cos (\beta x)}{x}dx &amp;= \int_{0}^{\infty}dx\int_{\alpha}^{\beta}\sin (yx)dy\\ &amp;=\int_{\alpha}^{\beta}dy\int_{0}^{\infty}\sin(yx)dx\\ &amp; \\ &amp;   \qquad\text{let $ yx=u $}\\ &amp; \\ &amp;=\int_{\alpha}^{\beta}\frac{1}{y}dy\int_0^{\infty}\sin u du\\ &amp;=\int_{\alpha}^{\beta}\frac{1}{y}dy\left( -\cos u|_{\infty}+\cos u|_0 \right)\\ &amp;=\log\frac{\beta}{\alpha}(-\cos(\infty)+1)\\ &amp;=\log\frac{\beta}{\alpha}-\cos(\infty)\log\frac{\beta}{\alpha} \end{align}$$  But $ \cos(\infty) $ does not exist right? Does it mean the integral actually diverse?  Edit: The question comes from   Who can point out my mistake in the above deduction? Render looks darker than Look Dev mode I have been texture painting in look dev mode. But when i try to render it, everything looks very dark. I have 3 sun lights in my scene, ambient turned on in Eevee settings. What am i missing? I really like the look of my scene in Look Dev mode, how can i achieve the same look when my scene is rendered too? How to register domain for 15 years? I have heard that Google trust that website which has long expiry date. that want to know how can I register my domain for a long period. Why does a star beyond a certain mass limit (Chandrasekhar limit) only become a black hole? Why does a star beyond a certain mass limit () only become a black hole?   A star is first made of hydrogen, it undergoes nuclear fussion reaction combining into helium and releasing a large amount of energy. This process continues till star is made up of iron core as iron has largest value of binding energy per nucleon, after this if mass of the star is above the value of Chandrasekhar limit it becomes a black star, what is the reason for this and why is certain mass limit required? Can you give us some way to access pending Reviews for a post from the post itself?  I think this would also help improve the accuracy and quality of reviews done, as users that come across the post naturally are more likely to be better qualified to evaluate the post, and are already taking the time to understand the post's content anyways.  In most cases, these users will be able to provide better reviews than those who are  [often quickly] going through the Review Queues.  In addition, I think it would help get more reviews done, as someone viewing a post through normal site usage would be more likely to take an interest in helping maintain that post. I know I definitely would be more active with this instead of having to go through the massive Review Queue. The task of helping fix one question that I am currently reading is far more do-able than cleaning up a large queue of questions that is largely full of content I don't care about.  I'm thinking of something like a reviews(1) link that is accessible from the post itself that appears based on your rep, or perhaps even on your rep within the post's tags. I was wondering how to run a command as another user from a script.  I have the script's owner set as root. I also have the following command being run within the script to run the command as the hudson user:  su -c command hudson   Is this the correct syntax? We all know that a simple statement such as:  SELECT * FROM stuff;   should not produce an ordered result. However, when I try to demonstrate this, it always comes out in primary key order.  Moreover a statement such as:  SELECT thing,whatever FROM stuff GROUP BY thing,whatever;   always appears to sort thing by the last field in the GROUP BY clause, which is not at all helpful.  The question is, under what circumstances does SQL SERVER order the result even if unasked, and what can I do to stop this?  I am trying to demonstrate to my students that order is indeterminate unless specified, but this is not helping my case.  I admit that I am working with a small set of sample data.  Thanks Where do executables look for shared objects at runtime? I understand how to define include shared objects at linking/compile time. However, I still wonder how do executables look for the shared object (*.so libraries) at execution time.  For instance, my app a.out calls functions defined in the lib.so library. After compiling, I move lib.so to a new directory in my $HOME.  How can I tell a.out to go look for it there? What is a simple example of a limit in the real world? I've got an iPhone app with icon file Icon.png.  This icon shows up properly when the app is on the phone itself, but it doesn't show up in the applications pane in iTunes.  What do I need to do to get it to show up properly? Dual monitor setup, why does NVidia 340 Prime only detect one monitor? I know that Wubi is slower in disk IO, but I can't find anything about it online other than vague statements such as "slightly slower disk seeks".  Has anyone done any benchmarks to quantify the speed difference?