# The Difference Between Log and Ln

1. Dec 5, 2014

### basty

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

2. Dec 5, 2014

### Staff: Mentor

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.

3. Dec 5, 2014

### Maged Saeed

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

4. Dec 5, 2014

### 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)".

Last edited by a moderator: Dec 5, 2014
5. Dec 10, 2014

### HakimPhilo

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

6. Dec 10, 2014

### pasmith

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

7. Dec 10, 2014

### Staff: Mentor

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.

8. Dec 10, 2014

### axmls

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