We all have heard about the natural logarithm for any number. Basically we all know that the natural logarithm is the logarithm to the base of $e$,which is a transcendental number. Now what about the other transcendental numbers such as $\pi$ . How would the logarithm to the base of $\pi$ behave. By behave I mean how will the graph of this kind of a logarithm look like,what will be its properties?

Thank you


I think you'll find that all logarithmic curves 'look' and behave the same. They only differ by a scalar factor. For example:

$$\text{If } y = \pi^x\text{, then we want}\log_\pi(\cdot)\text{ such that }x = \log_\pi{y}$$ $$\text{But }y = \pi^x = e^{\ln(\pi)x}\text{, so } x = \frac{\ln(y)}{\ln(\pi)}$$ Putting this together: $$\log_\pi(x) = \frac{\ln(y)}{\ln(\pi)}$$ Where $\ln(x)$ is the 'natural' logarithm. This property hold for all Real $c>0$ as an exponential base.

  • $\begingroup$ Technically you need positive $c$. $\endgroup$ – Ian Mar 14 '15 at 17:59
  • $\begingroup$ @jameselmore but what if the base is not real for eg $i$ then what will happen to the logarithm $\endgroup$ – user210387 Mar 15 '15 at 4:21
  • $\begingroup$ @SayanChattopadhyay, I'm afraid I do not have enough experience with the logarithm of a complex number to be able to give an accurate explanation. I can imagine that the result will be similar to the one above, the question becomes what corrections or modifications are needed to the $\ln$ function such that we can calculate $\ln(i)$ $\endgroup$ – jameselmore Mar 16 '15 at 17:07
  • $\begingroup$ Seeing as I answered your original question, could you kindly accept this answer? $\endgroup$ – jameselmore Mar 16 '15 at 18:03

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