Editing 1162: Log Scale

{{comic
| number    = 1162
| date      = January 18, 2013
| title     = Log Scale
| image     = log scale.png
| titletext = Knuth Paper-Stack Notation: Write down the number on pages. Stack them. If the stack is too tall to fit in the room, write down the number of pages it would take to write down the number. THAT number won't fit in the room? Repeat. When a stack fits, write the number of iterations on a card. Pin it to the stack.
}}

==Explanation==
This comic is a [[:Category:Tips|tip]], specifically the first [[:Category:Science tip|''science'' tip]]. As with most of Randall's tips, it is technically interesting for some applications but not very practical. Uranium is stated to have 76 million MJ/kg, while the next highest material shown on the graph (gasoline) has 46 MJ/kg. Thus the uranium graph should be taller by a factor of 76,000,000/46 = 1.652 million. So, if the gasoline graph were 9mm in height, the uranium graph should be a bit more than 14.868 million mm tall, or nearly 15&nbsp;km (9.2 miles) tall. Thus the need to fold the paper.

It should be noted that the method of extracting energy from the first 4 materials ({{w|combustion}}) is completely different from the method used with uranium ({{w|nuclear fission}}). If the technology existed to use {{w|nuclear fusion}} at the time of the comic, then the first 4 materials would yield a higher energy density than uranium.

A {{w|Logarithmic scale|log scale}} is a way of showing largely unequal data sizes in a comprehensible way, using an exponential function between each notch on the y axis of a graph. So for example the first on a Y axis of a graph using a log-10-scale would be 1, then 10, then 100 and 1000 for the fourth. A {{w|logarithm|log/logarithmic function}} is the {{w|inverse function|inverse}} of a corresponding {{w|Exponential growth|exponential function}}. A log-scale version of the chart in the comic would look like the image on the left.

[[File:Log_Chart_1162.png|500px|thumb|right]]

The log scale can also be abused to make data look more uniform than it really is. On a log scale the energy density of uranium looks larger than that of the other materials, but not dramatically so. The joke is that if one wanted to make their point "properly," they would go ahead and use ridiculous amounts of paper to show the difference between bars using a linear scale; this method would focus more on the shock factor of the differences in question, and less on actual communication/representation of data. Cueball seems to be passionate about the MJ/kg of uranium, so he would rather demonstrate the grandeur of the data than use a more efficient scale.

See {{w|Logarithmic scale#Common uses|these examples}} for well known day-to-day measurements which are measured on a log-scale.

The title text mentions computer scientist [[Donald Knuth]]; the fictional notation is a parody of {{w|Knuth's up-arrow notation}}. Using paper thickness as the basis for a log scale would probably give the exponential function a very large base. However, it can be noted that Knuth's up-arrow notation can handle numbers far, far larger than this paper stack notation; for example the number 3↑↑↑3, also known as Tritri<ref>https://googology.wikia.org/wiki/Tritri</ref>, very compact in up-arrow notation, would require a number of iterations pinned to the stack on the order of several trillion. 3↑↑↑↑3 , also known as Grahal<ref> https://googology.wikia.org/wiki/Grahal </ref>, would require a number of iterations that is not only too large to write down, but attempting to write that number using the same paper stack notation would require printing off a ''second'' stack of several trillion iterations just to hold the ''number'' pinned to the first stack. By repeating this multi-stack repetition, you reach the limit of up-arrow notation.

It should be noted that Randall has used log scales in past comics.

==Transcript==
:[A bar chart on a piece of paper, with a second piece of paper attached to it.]
:[Title of the bar chart] fuel energy density of different materials in megajoules/kg
:[Values of the first 4 bars on the paper] 19 24 39 46
:[The different bars for Sugar, Coal Fat and Gasoline and Uranium on a linear scale with the bar for Uranium exceeding on the attached stack of paper]
:[Labels of the 5 bars on the paper] Sugar Coal Fat Gasoline Uranium
:[The uranium bar on the chart goes off the page onto a huge strip of paper folded up into a stack slightly taller than Cueball.]
:[Value on the top end of the paper strip] 76 000 000
:[Caption below the panel:]
:Science Tip: Log scales are for quitters who can't find enough paper to make their point ''properly''.

==Trivia==
This comic was seen in the ''[[What If? (book)|What If?]]'' book, taken from "a certain webcomic".

{{comic discussion}}
[[Category:Comics featuring Cueball]]
[[Category:Bar charts]]
[[Category:Physics]]
[[Category:Statistics]]
[[Category:Science tip]]
[[Category:Comics featuring Donald Knuth]]
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