The Difference Between Log and Ln

In summary, the main difference between log and ln is the base being used. Log typically means log10, while ln always means loge. However, in some computer science texts, log may refer to log2. The common logarithm (log10) is used because of our base 10 number system, while the natural logarithm (ln) has useful properties in calculus. With the invention of calculators, the use of common logarithms has decreased and log is now becoming a more common notation for the natural logarithm. In higher math, "log" often refers to the natural logarithm, but it is important to pay attention to context to determine the intended meaning.
  • #1
basty
95
0
Both are logarithms, what is the difference between log and ln?
 
Physics news on Phys.org
  • #2
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.
 
  • #3
$$lnx=log_ex$$
so that:
$$ln(e)=1$$
and
$$log(x)=log_{10}x$$
so that
$$log(10)=1$$
 
  • #4
They way you are using it, log(x), or "common logarithm" is the inverse function to [tex]10^x[/tex]. That is, if [tex]y= log(x)= log_{10}(x)[/tex] then [tex]x= 10^y[/tex]. ln(x), the "natural logarithm", is the inverse function to [tex]e^x[/tex] ("e" is an irrational number, approximately 2.718...). If [tex]y= ln(x)[/tex] then [tex]x= e^y[/tex]. The common logarithm is used because our number system is base 10 so it is relatively easy to tabulate: [tex]log_{10}(3.00\times 10^5)= 5+ log_{10}(3.00)[/tex] 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 [tex]y= e^x[/tex] the "instantaneous rate of change" of y, as x changes, is again [tex]e^x[/tex] 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:
  • #5
There's also ##\lg## which denotes ##\log_2##.
 
  • #6
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.
 
  • #7
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
From what I understand, in higher maths, "log" denotes the natural logarithm quite frequently. It should usually be obvious from the context.
 

1. What is the difference between log and ln?

The main difference between log and ln is the base of the logarithm. Logarithm with base 10 is denoted as log, while logarithm with base e (Euler's number) is denoted as ln.

2. How do you convert log to ln?

To convert log to ln, you can use the change of base formula: ln(x) = log(x) / log(e), where e is the base of the natural logarithm.

3. What is the significance of using logarithms in scientific calculations?

Logarithms are useful in scientific calculations because they allow for large numbers to be expressed in a more manageable way. They also help in simplifying complex equations and reducing the number of calculations needed.

4. Can log and ln be used interchangeably?

No, log and ln cannot be used interchangeably. While they both represent logarithmic functions, they have different bases and therefore, different results. It is important to pay attention to which base is being used when using logarithms in calculations.

5. What are some real-life applications of logarithms and natural logarithms?

Logarithms and natural logarithms have many real-life applications, including measuring the intensity of earthquakes, predicting population growth, and calculating the pH level of a substance. They are also commonly used in finance, computer science, and physics.

Similar threads

Replies
14
Views
613
Replies
3
Views
1K
  • Calculus
Replies
7
Views
2K
  • Calculus
Replies
4
Views
2K
Replies
14
Views
1K
Replies
12
Views
3K
  • Introductory Physics Homework Help
Replies
12
Views
563
Replies
5
Views
1K
Replies
4
Views
695
Back
Top