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I'm understanding logarithmic scales and have questions. The axis coordinate for a number $x$ on the scale is $log_{10}(x) = y$. This can then be used to create the tick marks where $x$ is the tick mark and $y$ is the coordinate on the scale as follows:

$T = \{(x,y) | (x = 10^p*l) \land (1 < l \le 9) \land l,p \in \mathbb{N} \land (y = log_{10}(x)) \}$

Something that mislead me, is that the tick marks are labeled by $b^p$ and not $p$ as I would have expected.

QUESTIONS

  1. Does the description of coordinates and tick marks, $T$, look correct?

  2. This page and the following image explains that if the distance $p$ between two coordinates $a = (x_0, y_0), b = (x_1, y_1)$ is $d = y_1 - y_0$ and the factor between the tick marks $f = x_1 - x_0$, then $nf = nd$ and $\frac f n = \frac d n$. What is happening here algebraically and in relation to logarithm?

  3. This page discusses how the geometric mean $\sqrt(a*b)$ is used to find the value between two tick marks. It's my understanding it's used because the scale is logarithmic and a standard mean cannot be used. Does anyone know the details on why the geometric mean can be use?

Quest 1

Quest 2

Nick
  • 1,071
  • A normal scale is additive: each new tick mark adds to your total. A logarithmic scale is multiplicative: each new tick mark multiplies by the given base. The geometric mean is the "multiplicative halfway point" between two numbers, whereas on a normal scale, you use the usual arithmetic mean to get the "additive halfway point". – Théophile Feb 22 '21 at 23:26
  • I think I understand the tick mark multiplication as posted earlier in the set theoretic form. Where I'm confused is why $log(2) - log(1) = log(4) - log(2)$. Is this an identity? $log_b(\frac 2 1) = log_b(\frac 4 2)$ apply $b^x$ on both sides is $\frac 2 1 = \frac 4 2 = 2$. I think thats it, but I still am confused why. – Nick Feb 23 '21 at 00:27

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