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program to illustrate how log(exp()) is linear even if bases differ

 Here’s a potential assignment for the gap between the lecture on logs and exponents and before the lecture on e.  In that lecture, we would also introduce that there is a number called “e” that is sufficiently useful to be built into the calculator, and two sets of exponential and log functions built in.  This assignment would have the students observe how the log of an exponential of a linear is always linear, even if the log and exponential are of different bases.  My hope is that students would be almost prepared to figure this out after your lecture.  Some will just observe, and others may actually figure it out – and the next lecture could explain, generalize, and then dig into why e is so interesting.  The following is dramatically linear:  They would use multiplication to compute the exponential directly by multiplying, repeatedly – say by 2 - and then plot the logs in both base 10 and base e.  The assignment would include characterizing the plot of the log-of-exponential-of-another-base - and ask if they are related and why.  In python:  p = 1 for i in range(100):   p = p * 2.0   r.set((i,log(p)))  r.set((i,log10(p)))   I don’t know if we should give them the program or not.  We also should ask if the result would be be similar if we multiplied by a constant different form 2.  In order to subvert common misconceptions, maybe we also should ask them to figure out whether there is proportionality constant among distinct exponentials (no), logs (yes) and logs of different exponentials (yes)