What does $\ln(\ln(x))$ simplify to? What does $\ln(\ln(x))$ simplify to?
First instinct was to write $\ln^2x$, but that looked odd, because $\ln(\ln(x))$ is not the same as $\ln(x)\ln(x)$ or $(\ln(x))^2$. 
Or is it ?
 A: Refer to the following figure:

The blue curve is $y = \log x$ (or $\ln x$ for those of you who like to use "ln" instead of "log").  The red curve is $y = (\log x)^2$, which is sometimes written as $\log^2 x$, similar to how it is customary to write $\sin^2 x$ for $(\sin x)^2$.  The yellow curve is $y = \log(\log x)$, which does not simplify, and there is generally no standardized shorthand for this expression, which we can regard as a composition of the logarithm with itself.  Sometimes, when repeated composition is desired, it might be written as $$f \circ f \circ \cdots \circ f(x) = f^n(x)$$ but this is very confusing, since this is not what we mean when we write $\sin^n x$, as $\sin(\sin(\sin(\cdots(\sin x)\cdots)))$ is quite a different thing.
Some authors might use alternative notations; e.g., $f^{(n)}(x)$ or $f^{[n]}(x)$ to denote $n$-fold compositions, but this could also be confused with higher-order derivatives.  Bottom line:  it is always better to briefly define your notation clearly if there is any doubt as to whether it is standard usage, since it's better to be understood if a bit wordy (like this post) instead of terse and opaque.
A: $\ln^2 x$ ought to mean $\ln\ln x$ but some people write $\ln^2 x$ when they mean $(\ln x)^2$.  I would avoid the notation unless I explain at the outset what I mean by it.  In either case it does not mean $\ln(x^2)$, which is the same as $2\ln x$.
Gauss wrote that $\sin^2 x$ ought to mean $\sin\sin x$ rather than $(\sin x)^2$.  But the latter meaning has become standard.
A: It doesn't simplify to anything nice. $\ln(\ln (x))$ is the composition of $\ln (x)$ with itself.
The rule you're talking about is  $\ln x+\ln y = \ln (xy)$.
Also, most of the times I've seen "$\ln^2(x)$", this is meant to denote $\ln(x)\cdot\ln(x)=(\ln(x))^2$.
A: $\ln(\ln(x))$ does not, in fact, simplify. You can write it as "$\ln^2(x)$" - I think that would be understood the right way - but beyond that, there's nothing really you can do. (Note that indeed $\ln(\ln(x))\not=(\ln(x))^2$ - for example, $\ln(\ln(e))=\ln(1)=0$, but $(\ln(e))^2=1$.)
A: $ln(ln(x))$ is indeed not the same as $ln(x)*ln(x)$, but it is the same as $(ln \circ ln)(x)$ where $\circ$ denotes composition. Composition is also a kind of multiplication (see group theory) so it makes sense to write it as $ln^2$. Just always check to what kind of a multiplication an exponent refers.
