...essentially, why would someone want to look at a logarithmic scale graph? What can be derived from a logarithmic scale graph that makes it worthy of presentation in addition to a linear scale version?  At  there is a graph in "linear scale" (the default) and a tab with a "logarithmic" scale version.  Here they are:
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I am trying to gain a more intuitive feeling for the use of logarithms.   So, my questions: What do you use them for? Why were they invented? What are typical situations where one should think: "hey, let's take the $\log$!"?     Thanks for the great comments!      Here a short summary what we have so far:      History:      Logarithms were first published 1614   by John Napier   (mathematician/astronomer) . He needed   them for simplifying the   multiplication of large numbers.      Today’s uses:         In regression analysis: If you expect a predictor variable to follow   a power/exponential law, take the   corresponding logarithm to linearize   the relationship.   In finance to calculate compound interests.   Or more general: to calculate the time variable in growth/decay   functions. (Nuclear decay, biological   growth…)   In computer science to avoid underflow. (Cool trick! But seriously:   32-bit? Take a double ;-)   In the Prime Number Theorem   For handling very large/small numbers (pH, etc.)   Plotting (if your scale gets too large)
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Is the below brief explanation correct and complete (at least in context)? Can it be improved/simplified without getting technical? I am especially concerned about the last sentence: Is that reasoning absolutely clear to a layman?            The problem can be viewed as drawing the above graph without lifting your hand and without retracing a line. Observe that each dot is connected by three lines. Also observe that you have to draw a line to arrive at a dot, and you have to draw a line to leave that dot. Because each dot is connected by three lines, each must be visited twice. But it is impossible to visit each dot twice (even number of times) and end up with each dot connected with three lines (odd number of times).
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Why do we care about eigenvalues of graphs?  Of course, any novel question in mathematics is interesting, but there is an entire discipline of mathematics devoted to studying these eigenvalues, so they must be important.  I always assumed that spectral graph theory extends graph theory by providing tools to prove things we couldn't otherwise, somewhat like how representation theory extends finite group theory.  But most results I see in spectral graph theory seem to concern eigenvalues  not as means to an end, but as objects of interest in their own right.  I also considered practical value as motivation, e.g. using a given set of eigenvalues to put bounds on essential properties of graphs, such as maximum vertex degree.  But I can't imagine a situation in which I would have access to a graph's eigenvalues before I would know much more elementary information like maximum vertex degree.  (EDIT: for example, dtldarek points out that $\lambda_2$ is related to diameter, but then why would we need $\lambda_2$ when we already have diameter?  Is this somehow conceptually beneficial?)     So, what is the meaning of graph spectra intuitively?  And for what practical purposes are they used?  Why is finding the eigenvalues of a graph's adjacency/Laplacian matrices more than just a novel problem?
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I'm currently starting a course about Graph Theory, and I've been explained about the Havel &amp; Hakimi degree sequence. However, I'm not too sure about it yet. First of all, I want to know if I understand it correctly, so I'm going to try to explain what I think the theory proves.  First, I've shamelessly stolen the following statement from the presentation:    Okay, so what I want to ask is the following.   What does "is graphic" mean exactly. Does that mean it's a graph? I know stupid question, but I cannot think of a degree sequence that's not a graph... What's the practical use of this. What problems could you solve with this.
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What is the historical background for this term?   I cannot quite see what is integral about an integral, even if we go back to the viewing it as the area under a curve.  It seems to me a strange choice of word.
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Is there good example code or a test project for explaining the . There are a lot of explanation links, but I want to have some good example code to show others without reinventing the wheel.
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Very new LaTeX user here!   There are a ton of impressive graphs on this site. I, however, am trying to make a nice looking line on a simple graph by plotting two (x,y) coordinates: (8, 2) and (-2, -3).  Can somebody please help me with this? Here's what I had in mind:   I've found a lot of information on parabolic graphs and much more advanced stuff, but nothing on really simple graphs.   Here's what I made:  \begin{tikzpicture}     \draw [thin, gray, -&gt;] (0,-5) -- (0,5)      node [above, black] {$y$};       \draw [thin, gray, -&gt;] (-10,0) -- (10,0)        node [right, black] {$x$};                   \draw [draw=red,ultra thick] (8,2) -- (-2,-3);   \end{tikzpicture}   I spent a long time trying to figure this out and mad a pretty stupid looking graph (which is huge):
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I feel intimidated in learning TikZ. I was wondering if there are some nice materials (or just your replies) at different level:   for providing big picture, which is important to beginners to not lose direction during learning. for comprehensive reference.   Examples are important, but only examples are not always clear for understanding the big picture and organization.
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I was having some thought about what would be the syntax to make the end point evaluation of derivatives or integrals. Such as making the | with the two end points of evaluation on the top an bottom of the line. Any suggestions would be wonderful.
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In the log-log regression case,  $$\log(Y) = B_0 + B_1 \log(X) + U \&gt;,$$  can you show $B_1$ is the elasticity of $Y$ with respect to $X$, i.e., that $E_{yx} = \frac{dY}{dX}(\frac{Y}{X})$?
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I have no idea where to even start. WolframAlpha cant compute it either.  $$\int^1_0 \log^2(1-x) \log^2(x) \, dx$$  I think it can be done with series, but I am not sure, can someone help a little so I can get a start??  Thanks!
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I've found lots of resources that say this is a real number if it's not rational, but what is a real number, really?  I mean what is the definition of a real number?  If nothing else, anyone know of a resource where I could find out myself?  Thanks!
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If you look the at  there is a graph of the Euler phi function up to 1000:   Out of the plot you can see various trend lines that appear to be linear.  Q1) Are these lines actually lines? (ie do all the points that appear to be on one line actually all solutions to some linear eqn?)  Q2) Given that they're lines and not just something that looks a bit like lines can they be described? and what points are on them?   Q3) Are the points that don't appear to be on lines actually just the first points on lines that would only be 'visible' if we zoomed out and evaluated the Euler phi up to (say) 10000 or whatever number is large enough.
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If I have a level-log model, such as y= β0+β1log(x),  I know the interpretation is "If we increase x by one percent, we expect y to increase by (β1/100) units of y."  So, my questions are   If y is a percentage variable, what is the correct interpretation of the resulting regression? Is it still the same but has a unit of percentage instead of unit, so the interpretation becomes "If we increase x by one percent, we expect y to increase by (β1/100) percent of y"? Or we don’t have to divide β1 by 100, so the interpretation becomes "If we increase x by one percent, we expect y to increase by β1 percent of y"? Can I convert my variables (ex: interest rate and inflation rate) to logarithm if it has a unit of percentage? If it can be done, can I change my model into log-log model, although my variable y has a unit of percentage, so the model becomes  log(y)= β0+β1 log(x) ? What is the correct interpretation of this model?
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I was wondering if anybody would explain (or please point me to a nice reference) as to why the ``discrete Laplacian" on a graph is actually called a Laplacian. Namely, how is it related to the standard Laplacian on $\mathbb{R}^n$ ? Is there a sense in which the discrete Laplacian on say, a square lattice, would converge to the standard Laplacian as the lattice spacing tends to $0$ ?
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Sorry about asking such an elementary question, but I have been wondering about this exact definition for a while. What power of $n$ is $\log(n)$. I know that it is $n^\epsilon$ for a very small $\epsilon$, but what value is $\epsilon$ exactly?
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Hello i have the following graph:    I'm trying to figure out if it's planar or not. I think it is not planar but i can't find a subgraph that is a subdivision of K3,3 or K5 , to use the kuratowski theorem.  p.s. i can only redraw the graph to show that it is planar or prove that       its not planar with kuratowski theorem. NO other methods allowed.