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Given we have a function as follows (context is the Weber-Fechner law):

$$\frac{\Delta x}{x} = C \tag 1$$

Plotting here $\Delta x$ as a function of $x,$ this is clearly a linear function with a slope of $C.$

Now a log-transform of (1) results in:

$$\log\left(\frac{\Delta x} x\right) = \log(C) \tag 2 $$

and hence:

$$ \log\left(\Delta x\right) = \log(x) + \log(C) \tag 3$$

Now it is stated that plotting $\log(\Delta x)$ as a function of $\log(x)$ results in a linear function with slope $1$ and intercept $C.$ How can I see (analytically) that this is true?

Michael Hardy
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Pugl
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    First of all, it's not true: most likely you mean the intercept is $\log C.$ With this understanding, there's nothing additional to see because (3) explicitly exhibits a linear function with unit slope and intercept $\log C.$ – whuber Nov 22 '18 at 15:09
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    It seems there's nothing left to say, then. – whuber Nov 22 '18 at 15:35
  • You've completely solved the problem, but you don't seem to realize it. – Michael Hardy Nov 22 '18 at 15:56
  • No, I don't realize it, that is the point of asking. Is it really not possible to explain instead of making semi sarcastic comments? – Pugl Nov 22 '18 at 16:08
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    I don't think anyone is being sarcastic on purpose. They are just puzzled about what you're puzzled about. If you write $\log {\Delta}x =: v, \log x =: u, \log C =: b$ then you have $v = b + u$, which is linear in $u$ with unit slope: does that help? – Nick Cox Nov 22 '18 at 16:20
  • But in this link, again page 2 second last section, they say that the intercept is C - without redefinition - this is what is confusing me as it seems to me that the intercept must be log(C)? Or am I misunderstanding this because it is a log-log function? http://www.cns.nyu.edu/~msl/courses/0044/handouts/Weber.pdf – Pugl Nov 22 '18 at 16:23
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    The source uses different notation, but you're right and the source is inconsistent. But watch out: many writers would feel free to think or imply that notation for uninteresting constants is defined informally and locally. Other way round, it's often considered pedantic and/or unnecessary to introduce a new symbol for something used just once or to insist that a meaning has changed. – Nick Cox Nov 22 '18 at 16:40
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    I learned this the hard way in 1972 when I spent about a week trying to understand one short paper; the last time I read it it took about 10 minutes because I then knew the idea inside out; and I knew that the author was extremely undisciplined about notation and that -- undeservedly -- journal reviewers and journal editor just assumed that he knew exactly what he meant. – Nick Cox Nov 22 '18 at 16:41
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    yes exactly, but as a beginner you don't know, and feedback like yours can be much more helpful than a "it is obvious". So, many thanks:)! – Pugl Nov 22 '18 at 16:52

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