# Properties of a different kind of a logarithm

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

$$\text{If } y = \pi^x\text{, then we want}\log_\pi(\cdot)\text{ such that }x = \log_\pi{y}$$ $$\text{But }y = \pi^x = e^{\ln(\pi)x}\text{, so } x = \frac{\ln(y)}{\ln(\pi)}$$ 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.
• Technically you need positive $c$. – Ian Mar 14 '15 at 17:59
• @jameselmore but what if the base is not real for eg $i$ then what will happen to the logarithm – user210387 Mar 15 '15 at 4:21
• @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 $\ln(i)$ – jameselmore Mar 16 '15 at 17:07