The Difference Between Log and Ln

[basty]

Both are logarithms, what is the difference between log and ln?

 

[Mark44]

Both are logarithms, what is the difference between log and ln?

The base being used. log usually (but not always) means log₁₀. ln always means log_e.

Occasionally, in computer science texts, log is used to mean log₂.

 

[Maged Saeed]

$$lnx=log_ex$$
so that:
$$ln(e)=1$$
and
$$log(x)=log_{10}x$$
so that
$$log(10)=1$$

 

[HallsofIvy]

They way you are using it, log(x), or "common logarithm" is the inverse function to $$10^x$$. That is, if $$y= log(x)= log_{10}(x)$$ then $$x= 10^y$$. ln(x), the "natural logarithm", is the inverse function to $$e^x$$ ("e" is an irrational number, approximately 2.718...). If $$y= ln(x)$$ then $$x= e^y$$. The common logarithm is used because our number system is base 10 so it is relatively easy to tabulate: $$log_{10}(3.00\times 10^5)= 5+ log_{10}(3.00)$$ so that it is sufficient to tabulate logarithms for 1 to 10.

While "e" is a rather peculiar number, it has some nice "Calculus" properties. For example, if $$y= e^x$$ the "instantaneous rate of change" of y, as x changes, is again $$e^x$$ which means that the "instantaneous rate of change" of ln(x) is 1/x, a very easy function. Since the invention of "calculators", common logarithms are used a lot less so that it is becoming common to use "log(x)" to mean the "natural logarithm" as well as "ln(x)".

 

[HakimPhilo]

There's also $\lg$ which denotes $\log_2$.

 

[pasmith]

There's also $\lg$ which denotes $\log_2$.

I'm sure I've seen "lg" used for base 10 logarithm in texts where "log" means natural logarithm.

 

[Mark44]

As far as I can tell, there's no hard and fast rule. Depending on context, I have see "log" used as log₁₀, ln, or log₂.

 

[axmls]

From what I understand, in higher maths, "log" denotes the natural logarithm quite frequently. It should usually be obvious from the context.