Derivative of Exponential Functions with Other Logs

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Discussion Overview

The discussion revolves around the differentiation of exponential functions using logarithms of bases other than e, such as base 10 or base 2. Participants explore the implications of using different bases in the context of calculus, particularly focusing on the chain rule and the change of base formula.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant inquires whether the derivative of an exponential function can be calculated using logarithms of bases other than e.
  • Another participant requests an example to clarify the initial question.
  • It is suggested that while it is possible to differentiate using other bases, an extra constant will appear due to the chain rule. The change of base formula for logarithms can be utilized to facilitate this process.
  • A participant emphasizes that the base e is unique because it results in a derivative of 1 after differentiation.
  • A follow-up question asks for further elaboration on the differentiation process, specifically how to apply logarithms of base 2 in the context of a given function.

Areas of Agreement / Disagreement

Participants generally agree that it is possible to differentiate exponential functions using other bases, but there are nuances regarding the appearance of constants and the application of the chain rule. The discussion remains unresolved regarding the specific steps and implications of using different bases.

Contextual Notes

Limitations include the need for clarity on the application of the change of base formula and the handling of constants that arise from the chain rule. The discussion does not resolve how to proceed with the differentiation process using specific examples.

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Can the derivative of an exponential function be calculated with logs base something other than e? Like base 10 or 2?
 
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An example of what you're trying to do would be welcome.
 
You can, but you will get an extra constant from the chain rule. Just use a change of base formula for log or rewrite an exponential in some other base and the term comes right out.

##e## is important precisely because it's the only base of the exponential where after differentiating the constant is ##1##.
 
theorem4.5.9 said:
You can, but you will get an extra constant from the chain rule. Just use a change of base formula for log or rewrite an exponential in some other base and the term comes right out.

##e## is important precisely because it's the only base of the exponential where after differentiating the constant is ##1##.

Could you maybe expand on that? Or point out somewhere where I could read about it?

If I start with "f(x+Δx) - f(x) = a^(x+Δx) - a^x" how do I continue with log_2 or anything else from here? Would I even go to that being equal to "a^x * (a^Δx - 1)"?
 

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