Douglas Blumeyer's RTT How-To: Difference between revisions

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One way to think of this is that 12:19:28 is an excellent low integer approximation of log(2:3:5). That's a really compact way of saying that each of these sets of three numbers has the same ratio between each pair of them:

* <math>\frac{19}{12} = 1.583 ≈ \frac{log(3)}{log(2)} = 1.585</math>

* <math>\frac{19}{12} = 1.583 ≈ \frac{\log(3)}{\log(2)} = 1.585</math>

* <math>\frac{28}{12} = 2.333 ≈ \frac{log(5)}{log(2)} = 2.322</math>

* <math>\frac{28}{12} = 2.333 ≈ \frac{\log(5)}{\log(2)} = 2.322</math>

* <math>\frac{28}{19} = 1.474 ≈ \frac{log(5)}{log(3)} = 1.465</math>

* <math>\frac{28}{19} = 1.474 ≈ \frac{\log(5)}{\log(3)} = 1.465</math>

You may be more familiar with seeing the base specified for a logarithm, but in this case the base is irrelevant as long as you use the same base for both numbers. If you don't see why, try experimenting with different bases and see that the ratio comes out the same<ref>[https://en.wikipedia.org/wiki/List_of_logarithmic_identities This list of logarithmic identities] has been an excellent resource for me in getting my head around logarithmic thinking. As you can see there, <math>\frac{log_{10}{a}}{log_{10}{b}} = log_{b}{a}</math>, so the base doesn't matter; you could put anything in there — 10, 2, e — and it still reduces to <math>log_{b}{a}</math>.</ref>.

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 One way to think of this is that 12:19:28 is an excellent low integer approximation of log(2:3:5). That's a really compact way of saying that each of these sets of three numbers has the same ratio between each pair of them:
-* <span><math>\frac{19}{12} = 1.583 ≈ \frac{log(3)}{log(2)} = 1.585</math></span>
+* <span><math>\frac{19}{12} = 1.583 ≈ \frac{\log(3)}{\log(2)} = 1.585</math></span>
-* <span><math>\frac{28}{12} = 2.333 ≈ \frac{log(5)}{log(2)} = 2.322</math></span>
+* <span><math>\frac{28}{12} = 2.333 ≈ \frac{\log(5)}{\log(2)} = 2.322</math></span>
-* <span><math>\frac{28}{19} = 1.474 ≈ \frac{log(5)}{log(3)} = 1.465</math></span>
+* <span><math>\frac{28}{19} = 1.474 ≈ \frac{\log(5)}{\log(3)} = 1.465</math></span>
 You may be more familiar with seeing the base specified for a logarithm, but in this case the base is irrelevant as long as you use the same base for both numbers. If you don't see why, try experimenting with different bases and see that the ratio comes out the same<ref>[https://en.wikipedia.org/wiki/List_of_logarithmic_identities This list of logarithmic identities] has been an excellent resource for me in getting my head around logarithmic thinking. As you can see there, <span><math>\frac{log_{10}{a}}{log_{10}{b}} = log_{b}{a}</math></span>, so the base doesn't matter; you could put anything in there — 10, 2, e — and it still reduces to <span><math>log_{b}{a}</math></span>.</ref>.