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I am trying to use the following input


however the output is:

plotted function

How can I adjust it so the line goes down to the PlotRange mark specified. Or is this a limitation of Mathematica?

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Your question is not especially original, it's going to be closed in a while. I'd do this: Plot[{Re@#, Im@#}&@Log[x + 1], {x, -50, 50}, Evaluated -> True, PlotStyle -> Thick], PlotRange serves a different purpose than you seem to expect, see documentation pages. – Artes Aug 19 '13 at 9:46
I just tried that and got the following: Link To Graph – VikeStep Aug 19 '13 at 9:53
Does it make you wonder? If it does, start to learn complex functions, there are many helpful questions. – Artes Aug 19 '13 at 9:59
It's a problem of how Plot[] works. There is only one pixel per x-value, and the curve at x=-1 is so steep that it doesn't take up more than one pixel in width. Use this instead: ParametricPlot[{x, Log[x + 1]}, {x, -1, 3}, PlotRange -> {{-2, 3}, {-10, 2}}] – Volker Aug 19 '13 at 10:12
@SjoerdC.deVries This poor question is not even a bit clear what the OP wants, why not e.g. Plot[Log[x + 1], {x, -2, 50}, PlotRange -> {{-2, 50}, All}] – Artes Aug 19 '13 at 12:23
up vote 9 down vote accepted

As I mentioned in the comments, it's a problem of Plot[]. Mathematica won't draw more than one pixel per pixel column. So I'd use ParametricPlot instead, which makes it a little better, but still not perfect. The line becomes so steep that you can't tell the difference between it and a straight line, so you just extend it manually:

ParametricPlot[{{x, Log[x + 1]}, {-1, -x - 9}}, {x, -2, 50},
     PlotRange -> {{-2, 50}, {-50, 10}}, PlotStyle -> {Blue, Blue}]

enter image description here

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Works perfectly, I decided to use a method similar to this, I still used Plot, but I added Epilog->Line[{{-1,-4},{-1,50}}] – VikeStep Aug 19 '13 at 10:29
@user9053 I don't think your soulution is in any way similar to the one given here. – Sjoerd C. de Vries Aug 19 '13 at 12:59
what I'm saying is, is that he explained why it wasn't working and said that I should manually extend the line, so instead of manually extending it in ParametricPlot, I did it in Plot – VikeStep Aug 20 '13 at 7:21

This is what I'd do, the idea is to use ParametricPlot[{b^y, y}] for Log[b, x]. So we can use Solve to make it a little bit more general:

Block[{x, y}, 
 With[{notimportantoptions = Sequence[PlotStyle -> Thick, PlotRange -> {All, {-25, 6}}, 
                                      AspectRatio -> 1,  AxesStyle -> Arrowheads@.05,
                                      AxesOrigin -> {0, 0}, BaseStyle -> {18, Bold}]

  ParametricPlot[{x, y} /. Solve[y == 2 Log[15 + x], x, Reals][[1]], {y, -25, 6}, 
                 RegionFunction -> (# < 10 &), notimportantoptions, Evaluated -> True]

enter image description here

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