The Difference Between Log and Ln

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Logarithms are categorized primarily as common logarithms (log) and natural logarithms (ln), with log typically referring to base 10 and ln to base e. In some contexts, particularly in computer science, log may also denote base 2. The inverse functions are defined such that if y = log(x), then x = 10^y, while for y = ln(x), x = e^y. The natural logarithm has unique calculus properties, such as the instantaneous rate of change being 1/x. Due to the prevalence of calculators, the use of log for common logarithms has declined, leading to varying interpretations based on context.
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Both are logarithms, what is the difference between log and ln?
 
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basty said:
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 loge.

Occasionally, in computer science texts, log is used to mean log2.
 
$$lnx=log_ex$$
so that:
$$ln(e)=1$$
and
$$log(x)=log_{10}x$$
so that
$$log(10)=1$$
 
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)".
 
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There's also ##\lg## which denotes ##\log_2##.
 
HakimPhilo said:
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.
 
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.
 
From what I understand, in higher maths, "log" denotes the natural logarithm quite frequently. It should usually be obvious from the context.
 

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