Natural logarithms of numbers to a base

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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).