 David Troyer

2021-12-29

What is the difference between log and ln?
I was told that its David Clayton

Usually $\mathrm{log}\left(x\right)$ means the base 10 logarithm; it can, also be written as ${\mathrm{log}}_{10}\left(x\right)$.
${\mathrm{log}}_{10}\left(x\right)$ tells you what power you must raise 10 to obtain the number x.
${10}^{x}$ is its inverse.
$\mathrm{ln}\left(x\right)$ means the base e logarithm; it can, also be written as ${\mathrm{log}}_{e}\left(x\right)$
$\mathrm{ln}\left(x\right)$ tells you what power you must raise e to obtain the number x.
${e}^{x}$ is its inverse. chumants6g

$\mathrm{ln}$ is always $\mathrm{log}e$ , the logarithm base e .
When you specify the base you should always use $\mathrm{log}b$ , $\mathrm{ln}b$ doesn’t make sense.
When $\mathrm{log}$ is used without an explicit base, it’s usually one of the following options:
The base is $e$ ,
At some previous point, the author explicitly set that “all logs should be understood as base b”,
From the context, the base, if not e , is clear - and it will usually be 2, or, less often, 10.
It doesn’t matter. For instance, if the statement is that a function is $O\left(\mathrm{log}n\right)$ ,it’s an equivalent statement to say it’s $O\left({\mathrm{log}}_{b}n\right)$ for any base b , since the different log functions are just the same function scaled by a constant, i.e. ${\mathrm{log}}_{a}\left(x\right)=C{\mathrm{log}}_{b}\left(x\right)$ for a constant C that only depends on a and b (but not x ). Vasquez

Yes, big difference.
Log is used for base ten. You’ve probably seen an equation like this:
${10}^{x}=y$, where y is equal to 10 raised to the power x.
If you know y how do you find x?
You take the Log of the equation ${10}^{x}=y$
$\mathrm{log}\left({10}^{x}\right)=\mathrm{log}y$
$\mathrm{log}\left({10}^{x}\right)=x$
So
$x=\mathrm{log}y$
Ln is the natural log. The natural log is used to solve for a when you know b in the following equation:
${e}^{a}=b$, where e is approximately 2.718, this is the natural exponential.
to solve for a when you know y, take the natural log of $\left({e}^{a}=b\right)$
$\mathrm{ln}\left({e}^{a}\right)=\mathrm{ln}b$
$\mathrm{ln}\left({e}^{a}\right)=a$
So
$a=\mathrm{ln}b$
So how does Ln y compare to $\mathrm{log}y$
Let $y={10}^{c}={e}^{d}$

$10={e}^{2}.3026$
Then $y={10}^{c}=\left({e}^{2.3026}{\right)}^{c}={e}^{\left[}2.3026c\right]$
$y={e}^{d}={e}^{\left[}2.3026c\right]$
$d=2.3026c$
$\mathrm{ln}\left(y\right)=2.3026\left[\mathrm{log}\left(y\right)\right]$
$\mathrm{log}\left(y\right)=0.4343\left[\mathrm{ln}\left(y\right)\right]$
Hope this helps

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