Series involving Logs I'm trying to find the name of, and a good online reference to, a type of "logarithm series", e.g.

$$(1+x)^9 = \sum_{k=0}^{\infty} \frac{9^k\ln^k(1+x)}{k!} $$

I realise that this comes from $x^y \equiv \operatorname{e}^{y\ln x}$ and the Taylor series of the exponential function.

Wolfram-Alpha gives this as a potential series for $(1+x)^9$ alongside, for example, the binomial. I also remember reading about this type of "logarith series" some years ago. I've tried to Google it but, as you can probably imagine, it just lists the series of the natural log.
We do have Taylor's Series and Fourier Series and each is extremely useful.  There are some applications where one of them is of the most use; some where the other is; and a few where either would do.

Series in terms of powers of log(1+x) are a form of Taylor's series in which you expand f(x) in a Taylor's series, substitute log(x) for x in the series, and thus get an expansion for f(logx).  Lucian has pointed this out for the function in your example.

I think the reason you are not finding good references is that series like this are not used for general purposes as are the Taylor's and Fourier series.  Mathematicians develop analysis of ideas that are useful and widely applicable.  Since the Taylor's series include the kind of logarthmic series you are interested in, all the work that has been done in developing that theory applies to your interest.
We do have Taylor's Series and Fourier Series and each is extremely useful.  There are some applications where one of them is of the most use; some where the other is; and a few where either would do.

Series in terms of powers of log(1+x) are a form of Taylor's series in which you expand f(x) in a Taylor's series, substitute log(x) for x in the series, and thus get an expansion for f(logx).  Lucian has pointed this out for the function in your example.

I think the reason you are not finding good references is that series like this are not used for general purposes as are the Taylor's and Fourier series.  Mathematicians develop analysis of ideas that are useful and widely applicable.  Since the Taylor's series include the kind of logarthmic series you are interested in, all the work that has been done in developing that theory applies to your interest.