Natural logarithms of numbers to a base Is the following expression valid?
$\ln_c (x)$
Natural logarithm of $x$ to the base $c$.
I have never seen anything like it, a colleague just showed it to me today and we've spent most of the day arguing about whether it's valid or not, please help out.
 A: In general: $$\log_c(a)=b \to c^b=a$$
$\ln$ is used specifically when $c=e\approx2.71828$
Thus:
$$\ln(a)=b\to e^b=a$$
Hence, your notation is incorrect.
A: Sometimes there are little differences about the logarithm. For example, $\log$, $\ln$, $\lg$ are all used.
For example, the notation what I've learned in the high school, was this:


*

*$\ln$ means the $e$-based logarithm,

*$\lg$ the 10-based logarithm

*and $\log_x$ the x-based log.

*There was also $\text{lb}$, as "logaritmus binaris", the 2-based logarithm.


On this notation, $\ln_c$ is incorrect, but as there is not a really standardized one, we can suspect the meaning wanted to be $\log_c$.
Note, the important part of the Math is not the actually used notation, but what it means (or tries to mean).
A: You can write in general $\log_c(x)$ which is the logarithm to base $c$. You can further write  $\ln(x)$ which is the logarithm to base $e$ which is also called the natural logarithm (hence the abbreviation $\ln(x)$). The modes ${\rm lnc}(x)$ or ${\rm ln}_c(x)$ should not be used.  
