I need to compare $\log_4 5$ and $\log_5 6$.  I can estimate both numbers like $1.16$ and $1.11$. Then I took smallest fraction $\frac{8}{7}$ which is greater than $1.11$ and smaller than $1.16$ and proove two inequalities: $$\log_4 5 &gt; \frac{8}{7}$$ $$\frac{7}{8}\log_4 5 &gt; 1$$ $$\log_{4^8} 5^7 &gt; 1$$ $$\log_{65536} 78125 &gt; 1$$ and $$\log_5 6 &lt; \frac{8}{7}$$ $$\frac{7}{8}\log_5 6 &lt; 1$$ $$\log_{5^8} 6^7 &lt; 1$$ $$\log_{390625} 279936 &lt; 1$$ thats why I have $\log_5 6 &lt; \frac{8}{7} &lt; \log_4 5$.  But for proving I need estimation both logarithms (without this estimation I cannot find the fraction for comparing). Can you help me to find more clear solution (without graphs)
I realize that I lack any numerical intuition for logarithms. For example, when comparing two logarithms like $\log_78$ and $\log_89$, I have to use the change-of-base formula and calculate the values with a calculator in order to determine which of the two is larger.  Can anyone demonstrate an algebraic/analytic method to find out which of the two is larger without using a calculator?
Elements of order 2 in $D_{2n}$ Im new at this abstract algebra stuff and im not comfortable with the proofs techniques yet, so I have a question related to the elements of order $2$ in $D_{2n}$.  Problem:   Prove that $\{x\in D_{2n}|x^{2}=1\}$ is not a subgroup of $D_{2n}$ for $n\geq 3$.   By now, I know that all the elements of the form $sr^{k}$ has order $2$ because I tested by brute force haha, but my real question is how do I prove prove that. How do I have to proceed to prove that $sr^k$ has order 2.  Any help will be really appreciated, thanks so much &lt;3 Using Xpath With Default Namespace in C# I've got an XML document with a default namespace. I'm using a XPathNavigator to select a set of nodes using Xpath as follows:     XmlElement myXML = ...;   XPathNavigator navigator = myXML.CreateNavigator(); XPathNodeIterator result = navigator.Select("/outerelement/innerelement");   I am not getting any results back: I'm assuming this is because I am not specifying the namespace. How can I include the namespace in my select? Prove that lim $(\sqrt{n^2+n}-n) = \frac{1}{2}$ Here's the question: Prove that $\lim_{n \to \infty} (\sqrt{n^2+n}-n) = \frac{1}{2}.$  Here's my attempt at a solution, but for some reason, the $N$ that I arrive at is incorrect (I ran a computer program to test my solution against some test cases, and it spits out an error). Can anyone spot the error for me?  $\left|\sqrt{n^2+n}-n-\frac{1}{2}\right| &lt; \epsilon$  $\Rightarrow \left|\frac{n}{\sqrt{n^2+n}+n} - \frac{1}{2}\right| &lt; \epsilon$  $\Rightarrow \frac{1}{2}-\frac{1}{\sqrt{1+\frac{1}{n}}+1} &lt; \epsilon$  $\Rightarrow \frac{1}{\sqrt{1+\frac{1}{n}}+1} &gt; \frac{1}{2} - \epsilon = \frac{1-2 \epsilon}{2}$  $\Rightarrow \frac{1}{\sqrt{1+\frac{1}{n}}} &gt; \frac{1-2 \epsilon}{2}$  $\Rightarrow \frac{1}{\sqrt{\frac{1}{n}}} &gt; \frac{1-2 \epsilon}{2}$  $\Rightarrow \sqrt{n} &gt; \frac{1-2 \epsilon}{2}$  $\Rightarrow n &gt; \frac{4 {\epsilon}^2-4 \epsilon +1}{4}$ Why is vote time missing in the SE data dump and SEDE? Can a single PS Plus subscription be used on two different profiles on the same PS4? Both my girlfriend and I play on the PS4 at home, so there are 2 profiles on PS4. When we set our profile in PS4, we set the console as the primary console.  Now we want to buy PSN Plus subscription, and I want to know if the same subscription can be used in both profiles, or  only in one.   In case the same subscription can be used in both profiles, how can we set it up? Or is automatic? jQuery If DIV Doesn't Have Class "x" Where is Jon going? The positive integer solutions for $2^a+3^b=5^c$ What are the positive integer solutions to  the equation     $$2^a + 3^b = 5^c$$    Of course $(1,\space 1, \space 1)$ is a solution. New messages flagged blue bubble does not go away For every twice differentiable function $f : \bf R \rightarrow [–2, 2]$ with $(f(0))^2 + (f'(0))^2 = 85$, which of the following statements are TRUE? For every twice differentiable function $f : \mathbf R \rightarrow [–2, 2]$ with $(f(0))^2 + (f'(0))^2 = 85$, which of the following statement(s) is (are) TRUE ?  (A) There exist $r, s\in \bf R$, where $r &lt; s$, such that $f$ is one-one on the open interval $(r, s)$  (B) There exists $x_0 \in (–4, 0)$ such that $|f'(x_0)| &lt; 1$   (C) $\lim\limits_{x\to\infty}f(x) = 1$   (D) There exists $\alpha\in(–4, 4)$ such that $f(\alpha) + f''(\alpha) = 0$ and $f '(\alpha)\ne0$  My attempt is as follows:-  $A)$ If function is continuous, then definitely in some part of the interval it will be increasing or decreasing hence will be one-one on the open interval $(r,s)$   $f(x)$ cannot be a constant function as in that case $f'(x)=0$ which means $f(0)=\sqrt{85}$, but it doesn't belong to co-domain of $f$  B) By mean value theorem we can say that there exists some $c\in(-4,0)$ for which $4f'(c)=f(0)-f(-4)$  Suppose if $f(0)=2$ and $f(-4)=-2$, then we can only say that there exists $c$ such that $f'(c)=1$. It will not be necessary that there exists a $c$ for which $|f'(c)|&lt;1$  C) $\lim\limits_{x\to\infty}f(x) = 1$, this is not at all necessary.  D) If function is not constant in $(-4,4)$, then we can safely say that at some point $\alpha$, $f(\alpha)\ne0$  But how to know $f(\alpha)+f''(\alpha)=0$  Actual answer is $A,B,D$ but I am able to ascertain only option $A)$. Please help me in this. $\frac{1}{99989999} = 1.00010002000300050008001300210034005500890144... \times 10^{-8}$    (), which includes the Fibonacci sequence $(1\ 1\ 2\ 3\ 5\ 8\ 13\ 21\ 55\ 89\ 144\ldots )$.  This is fascinating to me but I can't figure out how this works, and a Google search doesn't seem to reveal anything. Random contacts on my iPhone 5 I recently bought a new iPhone 5. When I was going through my contacts today, I noticed a lot of email adresses stored along with the other numbers. I didn't even know most of them! What has happened? It is well known from Noether's Theorem how from continuous symmetries in the Lagrangian one gets a conserved charge which corresponds to linear momentum, angular momentum for translational and rotational symmetries and others.  Is there any elementary argument for why linear or angular momentum specifically (and not other conserved quantities) are conserved which does not require knowledge of Lagrangians? By elementary I mean, "if this is not so, then this unreasonable thing occurs".   Of course, we can say "if we want our laws to be the same at a different point in space then linear conservation must be conserved", but can we derive mathematically the expression for the conserved quantity without using the Lagrangian?  I want to explain to a friend why they are conserved but he doesn't have the background to understand the Lagrangian formalism. I am teaching how to convert from one base to another base, however, I can't find a way to do something like this:    I know little about Tikz, the only thing I have been able to produce is a simple square with a couple of lines over it, so I don't even know where to start. Any suggestions? I don't mind typing all the numbers, what I want to do is to automatize the process of creating the lines. Very large log files, what should I do? ( deals with a similar issue, but it talks about a rotated log file.)  Today I got a system message regarding very low /var space.   As usual I executed the commands in the line of sudo apt-get clean which improved the scenario only slightly. Then I deleted the rotated log files which again provided very little improvement.  Upon examination I find that some log files in the /var/log has grown up to be very huge ones. To be specific, ls -lSh /var/log gives,   total 28G -rw-r----- 1 syslog            adm      14G Aug 23 21:56 kern.log -rw-r----- 1 syslog            adm      14G Aug 23 21:56 syslog -rw-rw-r-- 1 root              utmp    390K Aug 23 21:47 wtmp -rw-r--r-- 1 root              root    287K Aug 23 21:42 dpkg.log -rw-rw-r-- 1 root              utmp    287K Aug 23 20:43 lastlog    As we can see, the first two are the offending ones. I am mildly surprised why such large files have not been rotated.  So, what should I do? Simply delete these files and then reboot? Or go for some more prudent steps?  I am using Ubuntu 14.04.  UPDATE 1  To begin with, the system is only several months old. I had to install the system from scratch couple of months back after a hard disk crash.  Now, as advised in ,  I first checked the offending log files using tail, no surprise there. Then, for deeper inspection, I executed this script from the .  for log in /var/log/{syslog,kern.log}; do    echo "${log} :"   sed -e 's/\[[^]]\+\]//' -e 's/.*[0-9]\{2\}:[0-9]\{2\}:[0-9]\{2\}//' ${log} \   | sort | uniq -c | sort -hr | head -10 done   The process took several hours.  The output was in the line of,   /var/log/syslog : 71209229  Rafid-Hamiz-Dell kernel:  sda3: rw=1, want=7638104968240336200, limit=1681522688 53929977  Rafid-Hamiz-Dell kernel:  attempt to access beyond end of device 17280298  Rafid-Hamiz-Dell kernel:  attempt to access beyond end of device    1639  Rafid-Hamiz-Dell kernel:  EXT4-fs warning (device sda3): ext4_end_bio:317: I/O error -5 writing to inode 6819258 (offset 0 size 4096 starting block 54763121030042024)        &lt;snipped&gt;  /var/log/kern.log.1 : 71210257  Rafid-Hamiz-Dell kernel:  attempt to access beyond end of device 71209212  Rafid-Hamiz-Dell kernel:  sda3: rw=1, want=7638104968240336200, limit=1681522688    1639  Rafid-Hamiz-Dell kernel:  EXT4-fs warning (device sda3): ext4_end_bio:317: I/O error -5 writing to inode 6819258 (offset 0 size 4096 starting block 954763121030042024)    (/dev/sda3 is  my home directory. As we can find,   lsblk /dev/sda NAME   MAJ:MIN RM   SIZE RO TYPE MOUNTPOINT sda      8:0    0 931.5G  0 disk  ├─sda1   8:1    0 122.1G  0 part / ├─sda2   8:2    0   7.6G  0 part [SWAP] └─sda3   8:3    0 801.8G  0 part /home    Why a process will want to write beyond the limit is actually outside the scope of my comprehension. Perhaps I will want to ask a different question in this forum if this continues even after a system update.)  Then, from  (you may want to check  for a deeper understanding), I executed,   sudo su - &gt; kern.log &gt; syslog   Now, these files have zero sizes. The system is running fine before and after a reboot.   I will watch these files (along with others) in the next few days and report back should they behave out-of-line.  As a final note, both  the offending files (kern.log and syslog), are set to be rotated, as inspection of the files (grep helped) inside  /etc/logrotate.d/ shows.  UPDATE 2  The log files are actually rotated. Looks like the large sizes were attained on a single day. I think this is a duplicate of an old question without a good answer:  My child was doing schoolwork on her school-issued Macbook Air. Suddenly lighter greys are totally white. This generally gives the images on the screen a vague feeling of being &quot;off&quot;, and pixellates all curved edges. The biggest problem is that text on some of her sites is unreadable. I've checked the following:  Color profiles - it looked like it was using the default, but a cycled through a few built-ins. I could see noticeable changes to color balance, but the light greys remained the same as white. Also deleted contents of ~/Library/ColorSync/. Accessibility - my child has accidentally inverted colors before, this isn't that. No other accessibility settings seemed related to color. Display settings - Modifying brightness was also fruitless. The screen overall got brighter or darker, but still couldn't distinguish between light grey and white Rebooting - colors look fine until shortly after logging in - seems connected to user profile somehow.  Here are some images to demonstrate the issue. This is a generic color test image:  On the Mac, it looks like this (simulated with photoshop):  Any solution would be appreciated, but if there is a keyboard shortcut that can cause this, it would be the most likely explanation (8 year old says they &quot;might have rested their arms on the keyboard&quot; during a class meeting) What are the benefits of nickel-plated chains? Bicycle chains with higher model numbers (and slightly higher prices) are often nickel-plated or half-nickel-plated. What are the benefits of nickel-plated chains over non-nickel-plated chains?  How durable is the plating--does it wear off, or do its benefits last for the life of the chain? Compute $\mathbb{P}\{ W_t < 0 \, \, \text{for all} \, \, 1 < t < 2\}$ for a Brownian motion $(W_t)_{t \geq 0}$ Let $(W_t)_{t \ge 0}$ be a standard one-dimensional Brownian motion. Calculate $$\mathbb{P}\{ W_t &lt; 0 \, \, \text{for all} \, \, 1 &lt; t &lt; 2\}.$$ I can only think that this will be conditional on $W_1$. Please tell how to proceed? Can a multi-core processor run multiple processes at the same time? I understand that a multi-core processor can run multiple threads of the same process at the same time using its different cores. Can it also run multiple processes at the same time(each core running a different process)? How do I identify which Linux distro is running? We have some new hardware in our office which runs its own customized Linux OS.   How do I go about figuring which distro it's based on?