Should the notation “$\log x$” mean the decimal logarithm? I'm confused how do i can justifying to my students the difference between the two Notation “$\log x$” and “$\ln x$” however i checked the definition here in wikipedia. Really many people understand that the notation “$\log x$” must be the decimal logarithm (base $10$). But what I think is $\log x$ can be written as $\log_{e}x$ which is $\ln x $. The difference occurs only when I put the base $a$ as $\log_{a}x$ different from $e$.
My question here is:

Should the notation  “$\log x$” mean the decimal logarithm and how can I give the correct notation with the correct definition to my students?

Thank you for any help 
 A: In pure mathematics there is one and only one logarithm $\log$ defined to be the inverse function of $\exp$. Personally I do not like the notation $\ln$ since for example in real, complex or functional analysis you will never encounter any other logarithm than the natural one. The most clear notation I've encountered in the Book Analysis I by Vladimir A. Zorich: He uses $\log$ to denote the natural one and $\log_a$ for $a>0$ is the inverse function of the exponential mapping $a^x := \exp(x\log a)$. See, even here we use just the most fundamental transcendental functions. 
But this is my way of how I would define it. There are several possibilities and most of the time definitions are a matter of taste. The most important part is to be consistent.
A: It honestly doesn't matter as long as you are clear. For example, wolfram uses $\log(x)$ even when it means the natural log but clarifies this in a footnote. I personally prefer $\ln(x)$ (when teaching highscool) because it requires no further clarification.
At the end of the day it doesn't really matter; definitions and notation are two separate things. Just pick one and be consistent. Don't penalize your students for choosing certain accepted notations over others as long as they are clear about their intent i.e. $\log_e(x)$ is as valid as $\ln(x)$ which is as valid as $\log(x)$ where $\log(x)$ is the natural logarithm. 
Being too picky about these things can cause students to associate Mathematics with some suffocating and obscure set of rules and rituals that really have nothing to do with Mathematical thinking.
A: Calculators denote the the logarithm in base 10 as "Log" and the natural logarithm by "Ln". Teachers then need to refer to the logarithms in the way these are indicated by the calculators as students are first exposed to simple pre-calculus problems involving exponentials and logarithms where they need to use their calculators, like powers of two and then trying to find for which power of two the outcome is some given number etc. etc.
When students are exposed to calculus, they will have gotten used to "Ln" for natural logarithm, so the teachers will not switch to the official notation. Only at University will students have to get used to "log" for the natural logarithm because that's the way it's used in the scientific literature.
