# How does one draw a logarithmic scale?

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• DaveC426913
In summary, the author is trying to create a logarithmic scale for distances, but is not sure how to do it. He finds a slide rule on eBay and uses it to approximate distances in millions of miles.
DaveC426913
Gold Member
TL;DR Summary
I don't know how to make a logarithmic ruler
So I'm playing with this visualization from this other thread

and I'm brute-forcing the "days" scale because don't really know how to place the marks.
(by brute-forcing, I mean I am using SUVAT to calculate the distance one can travel in one day, then redoing it to calc the distance in two days, then three, etc.) I don't really know how to shortcut that into measurements and I can neither extrapolate to longer durations, not interpolate to shorter dimensions. I would be nice to put some finer "hours" increments in there.

There's a way to do it formulaically for sure, but since I'm already doing geometry I suspect there's a geometric shortcut, yes? How can I draw a logarithmic scale?

Use a slide rule as a ruler?

Mark44
apostolosdt said:
Use a slide rule as a ruler?
I'm just a couple of years too young for those. I saw them around but I was never taught how to use one.
It would be a challenge to find one I think.

Are all logarithmic scales proportional? I mean, if I had one that went up an order of magnitude every inch, I could just scale it to two inches if I wanted. But -er- here's where I show my ignorance. That's log 10 right? What if the scale I'm working with is, say, log 3? I can't scale a log 3 by stretching a log 10, can I? Jeez, I sound like a grade school kid.

apostolosdt said:
Use a slide rule as a ruler?
The numbers marked on a slide rule are at positions that are the log (base-10) of the numbers.
1 is at 0
2 is at ##\log_{10}(2)## or about .301 units
3 is at ##\log_{10}(3)## or about .477 units
...
10 is at 1 unit.

There are lots of slide rules for sale out there. Here's one I found on eBay for about \$20 (US) - https://www.ebay.com/itm/3255695050...1291&msclkid=f35b1a12fa3416cd5f267eb6cea0cd92

I personally have 5 sliderules of various kinds, including one that belonged to my wife's father.

Last edited:
DaveC426913 said:
TL;DR Summary: I don't know how to make a logarithmic ruler
...
How can I draw a logarithmic scale?
Tell 'Excel' or 'Numbers' to scale a chart axis logarithmically?

Below are the planets and dwarf planets plotted by distance from the sun.

Although the above plot might be useful, I don't understand how one would use a logarithmic ruler.

Data for above plot:
 Planets AU Mercury 0.4 Venus 0.7 Earth 1.0 Mars 1.5 Ceres 2.8 Jupiter 5.2 Saturn 9.5 Uranus 19 Neptune 30 Pluto 40

FactChecker
This is the part I'm trying to render:

One day of travel at 1g gets you distance X.
Two days of travel at 1g gets you more than 2x. But how much more?

The SUVAT formula involves t2. How does that affect the distance of 1 versus the distance of 2?

Here's my numbers using this SUVAT calculator:

One day journey:
u=0, a=1g, t=12h
d= 5.7 million miles
(double that for the decel phase)
= 11.4 million miles in one day

Two day journey:
u=0, a=1g, t=24h
d= 22.7 million miles
(double that for the decel phase)
= 45.4 million miles in two days

Three day journey:
u=0, a=1g, t=36h
d= 51.2 million miles
(double that for the decel phase)
= 102 million miles in three days

Four day journey:
u=0, a=1g, t=48h
d= 91 million miles
(double that for the decel phase)
= 182 million miles in four days

OK, so, every doubling of duration results in a quadrupling of distance. Got it.

Easy for multiples of two.

What about 3 or 5? Or any non-two-multiple?

What about dividing each day into hours? (OK, so ... the 12 hour mark in each day should be 4:1 between midnights. If a day is 100pixels, then noon should be at ... the 20 pixel mark )

I've still got to work out the distances for the outer system, which is an order of magnitude larger.

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If ##days## are an exponential function of ##millionMiles##, then taking the logarithm of the ##days## should get you back to a standard x,y cartesian coordinate system with a straight line function graph of ##\log(days)## versus ##millionMiles##

DaveC426913 said:
This is the part I'm trying to render:
View attachment 328766
This is a not a logarithmic scale, it is a quadratic scale. Which should not be a surprise because it comes from a quadratic equation ## s = ut + \frac 12 a t^2 ##. Simply mark your ruler at 11.4 mm, 45.4 mm, 102 mm and 182 mm.
Think about ## \frac{11.4}1, \frac{45.4}4, \frac{102}9, \frac{182}{16} \dots ##

DaveC426913 said:
I've still got to work out the distances for the outer system, which is an order of magnitude larger.
Then you are going to need an order of magnitude larger ruler.

Sorry, not sure that helps me.Let me be more specific:

I think I know how to measure quarter days and half days:

Here's the scale for a one day trip. I want to put hours (or at least quarter days) in:

So let's say in one day they can travel, say, 100 pixels:

Doubling the duration quadruples the distance.
In a quarter day they can travel 6.25 pixels.
Double the duration to a half day and they can travel 4x as far, which will be 25 pixels.
Double the duration to a full day and they can travel 4x as far, which will be 100 pixels.
So I have 6h (quarter day), 12h (half day) and a full day.So how many pixels is three-fourths of a day?

pbuk said:
Then you are going to need an order of magnitude larger ruler.
No, I need an order of magnitude smaller scale.

DaveC426913 said:
Sorry, not sure that helps me.
Really?

DaveC426913 said:
So how many pixels is three-fourths of a day?
Fill in the gap: ## \frac{6.25}{1^2}, \frac{25}{2^2}, ???, \frac{100}{4^2} ##. Or if you prefer ## \frac{6.25}{6^2}, \frac{25}{12^2}, ???, \frac{100}{24^2} ##.

DaveC426913
pbuk said:
Really?
That was in response to FactChecker's post.

pbuk said:
Fill in the gap: ## \frac{6.25}{1^2}, \frac{25}{2^2}, ???, \frac{100}{4^2} ##. Or if you prefer ## \frac{6.25}{6^2}, \frac{25}{12^2}, ???, \frac{100}{24^2} ##.
Ohhhhhhh! That's was the piece I was missing!

So... (6.25x9) = 55 ...

pbuk
DaveC426913 said:
Ohhhhhhh! That's was the piece I was missing!
I think I could hear the switch click as the light bulb went on!

DaveC426913 said:
So... (6.25x9) = 55 ...
Well you might be better off with ## \frac {100}{16} 9 = 56.25 ##, but you get the idea. I'd just build a spreadsheet calculating ## s = 0.5 a t^2 ## for ## t = 1, 2, 3... ## and linearly scaling ## s ## as appropriate.

Wait, that doesn't work.

If this were the right smooth scaling of quarter days, then it should work just fine between 1 and 2 days by just a straight scaling up:

But it doesn't. The last quarter of day one is much larger than the first quarter of day two.

DaveC426913 said:
Wait, that doesn't work.

If this were the right smooth scaling of quarter days, then it should work just fine between 1 and 2 days by just a straight scaling up:
View attachment 328824
But it doesn't. The last quarter of day one is much larger than the first quarter of day two.
Between 0 and 1 you have ## 6^2, 12^2, 18^2 \text{ and } 24^2 ##, between 1 and 2 you should have ## 30^2, 36^2, 42^2 \text{ and } 48^2 ##.

Edit: you should also notice that the length of the quarters increases by a fixed amount between each quarter - that is because we have quadratic growth: the second differences are constant, see https://www.bbc.co.uk/bitesize/guides/zy82ng8/revision/4

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pbuk said:
Well you might be better off with ## \frac {100}{16} 9 = 56.25 ##,
Derp. That's me speed-calcing in my head.

pbuk said:
but you get the idea. I'd just build a spreadsheet calculating ## s = 0.5 a t^2 ## for ## t = 1, 2, 3... ## and linearly scaling ## s ## as appropriate.

One day is 52.px.
That final column is total-pixels-from-zero.

Phew!

pbuk
DaveC426913 said:
I can't scale a log 3 by stretching a log 10, can I?
I think you can; try stretching the log10 scale by a factor of log3(10) (2.0959)

apostolosdt said:
Use a slide rule as a ruler?
The KISS method, right?

In short, yes, although I never heard of the KISS trend before. Anyway, good point, Bruzote.

Get yourself a spreadsheet package like Excel or a graphics program like Kaleidagraph, and plot one or two points using arithmetic scale. Then have the software switch each of the axes to logarithmic scale. This only requires a click or two of the mouse. Then have the software put a line through the data.

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