How to Expand Log(a+b) for Best Fit Analysis?

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Discussion Overview

The discussion revolves around the expansion of the logarithmic expression log(a+b) for the purpose of best fit analysis in a mathematical context. Participants explore various methods of transforming equations to facilitate plotting and regression analysis.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant inquires about expanding log(a+b), expressing difficulty in finding an obvious method.
  • Another participant questions the desired outcome, suggesting that the expression is already in its simplest form.
  • A different participant mentions that while it may be useful to factor log(a+b) as log(a) + log(b+1) in some cases, generally, no simpler form exists.
  • A participant introduces a related equation, X^3 = (cY+d)^2, and seeks advice on plotting it as a straight line graph, indicating a need for transformation.
  • One suggestion is made to take the logarithm of Y and plot x against log(Y), although it is noted that this does not yield a straight line representation of the original relationship.
  • Another participant emphasizes that the equation cannot be plotted as a straight line in standard form and questions the necessity of transforming it into a y=mx+c format.
  • Further clarification is provided that while the logarithmic transformation may not yield a linear relationship, it could still assist in finding regression constants from a dataset.
  • Participants express uncertainty about the effectiveness of various transformations (log-log, log-linear, linear-log) in achieving a straight line from the given equation.
  • One participant seeks additional context regarding the problem, asking if there are multiple data points available for determining the constants c and d for best fit analysis.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best method for expanding log(a+b) or transforming the related equation for plotting. Multiple competing views and uncertainties remain regarding the effectiveness of various approaches.

Contextual Notes

There are limitations regarding the assumptions made about the relationships between variables and the specific context of the problem, which remain unresolved. The discussion does not clarify the mathematical steps necessary for transformations or the specific nature of the data points involved.

Saoist
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anyoen know how to expand this? i can't think of any obvious way...
 
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What kind of result are you looking for - functions of a and b separately? As it stands, it is as simple as possible.
 
There's not much you can do. In some cases, it's useful to factor it as \log a+\log(b+1), but in general there's nothing simpler than the way you wrote it.
 
i have a deceptively simple question you see:

X^3 = (cY+d)^2

where c and d are constants, with x and y the variables. how would you plot the 2 variables as a straight line graph. I'm having an idiocy attack and can only think "log it..."
 
Take the log of Y and graph x, log y.
 
that doesn't plot that relationship as a straight line though does it?

i was under impression you had to transform [said equation] into a y=mx+c type form
 
Saoist said:
that doesn't plot that relationship as a straight line though does it?
i was under impression you had to transform [said equation] into a y=mx+c type form

You can't plot things like x^3=y^2 as a straight line on a normal graph.
 
I didn't mean that would give you a formula, but if you had a set of data, you could find the regression by plotting x, log y. It's not the answer but it's a way to get it.
 
Saoist said:
that doesn't plot that relationship as a straight line though does it?
i was under impression you had to transform [said equation] into a y=mx+c type form

No, none of log-log, log-linear or linear-log will make that equation a straight line.

What's the full context of the problem, do you have a number (more than 2) of x,y points and you wish to find constants c and d that give the "best fit" in some particular sense?
 

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