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There was given some data, in which I have carry out a linearizing procedure, using either a power model or a exponential model. From my understanding, power models and exponential models are linearized virtually in the same way, except my stats professor said they were not. How am I able to differentiate between a power model and an exponential model?

Here is the data given to me:

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whuber
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Ryan Koo
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    The example suggests an interest in accelerating processes over time (say $t$) A power model $y = at^b$ is hard to apply here without an obvious origin. An exponential model is not. Thus $y =\exp(a + bt)$ can be applied easily (which doesn't make it a good model). Perhaps more to the point, there is a big difference between logging one variable and logging two, which is likely to be part of your professor's argument! (If these aren't the models you're thinking of, then do spell out what you are considering.) – Nick Cox Mar 21 '18 at 16:14
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    A principled, systematic, effective, and robust method is explained and illustrated (with code) in my answer at https://stats.stackexchange.com/a/35717/919. You will find it doesn't produce anything reasonable with the raw data (because the trend is so strong) and that it produces an excellent result using the logarithms of the transistors. But that would be obvious from plots of the data and their logarithms anyway. In fact, your table already provides a nice plot: the number of digits needed to express the transistors appears to increase linearly with time. – whuber Mar 21 '18 at 17:03

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Very briefly, a power model involves taking the logarithm of both the dependent and independent variable.  The slope from the bivariate regression will produce the power.  For an exponential model, you only take the logarithm of the dependent variable.  The slope from the regression will produce the multiplicative growth rate.

Happy to elaborate more if need be.

Nick Cox
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Gregg H
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  • Hey, so you said in your post that you only take the logarithm of the dependent variable for an exponential model, which got me confused. The textbook we are studying out of introduces the equation for linearization as log _y_ = log _a_ + _x_ log _b_ . But the text also introduces an equation of the form ln (y) = a + b(x). Is there a specific instance where one of the equations is most appropriate, or does it not matter? Thanks. – Ryan Koo Apr 03 '18 at 04:08
  • For the exponential model, $ln(y)=a+b(x)$, if you exponentiate both sides you get $y = \exp(a+bx)=e^a·(e^b)^x$. So, $y$ is an exponential expression. Don't have ready examples for each readily available. – Gregg H Apr 03 '18 at 04:51
  • Found a website that has an example of each: http://www.real-statistics.com/regression/power-regression/ (If you go to the bottom of this "lesson" there is a link for exponential regression.) – Gregg H Apr 03 '18 at 05:11
  • So, when using a natural log or logarithm, it does not matter which one you use as long as the logarithm is of base e? – Ryan Koo Apr 03 '18 at 05:19
  • No, it won't matter which log you use. Recall the change of base formula: $\log_b(x) = \frac{\log_c(x)}{\log_c(b)}$. So, if you use a different log, it is just a matter of dividing by a constant. – Gregg H Apr 03 '18 at 05:26
  • One more question, in the website with the example you gave, I noticed that in a power model, you take the natural logarithm of both sides, correct? And for the exponential model, you take the natural logarithm of the y variable. Would that also apply to regular logarithms or is there a different method specific for logs? – Ryan Koo Apr 03 '18 at 15:10
  • This would work with any base for the logarithms. – Gregg H Apr 03 '18 at 15:34