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.