# What is the difference between log and ln?

## I was told that it's basically the same thing and can be used interchangeably, but if it's the same, what is the point of having another one? To reword it, if log and ln is the same, why use ln over log and vice versa? When should I use log/ln? Thank you in advance.

Dec 4, 2017

Usually $\log \left(x\right)$ means the base 10 logarithm; it can, also be written as ${\log}_{10} \left(x\right)$.

${\log}_{10} \left(x\right)$ tells you what power you must raise 10 to obtain the number x.

${10}^{x}$ is its inverse.

$\ln \left(x\right)$ means the base e logarithm; it can, also be written as ${\log}_{e} \left(x\right)$.

$\ln \left(x\right)$ tells you what power you must raise e to obtain the number x.

${e}^{x}$ is its inverse.

Dec 4, 2017

They are not the same!! They are both logarithms, but they are different logarithms.

#### Explanation:

There's a huge difference between log and ln!

A logarithm is a form of math used to help solve the following sort of problems:

${a}^{x} = b$

The question you're asking here is to what power do I need to raise $a$ to get $b$? This exact thing can be said using logarithms (as shown below):

${\log}_{a} b = x$

The relationship between logarithms and exponents is described below:

That value $a$ there is what we call our base, and it can vary based on what problem you're trying to solve.

When you have a base 10, then it's convention to just drop the base from the notation, since it's implied that you're talking about a base of 10.

So $\log \left(3\right)$ and ${\log}_{10} \left(3\right)$ are one and the same thing, the same way $x$ and $1 x$ are the same thing: they tell you the same thing, but one has superfluous information.

When you have a base $e$, you switch to $\ln$, and again drop the base from your notation.

So $\ln \left(3\right)$ is the exact same thing as ${\log}_{e} \left(3\right)$ .

As you can see, $\log \left(x\right)$ and $\ln \left(x\right)$ are not the same thing! They involve the same concept, and are both logarithms, but they are still different things.