Unravelling the Mystery of logn Identities

In summary, the first identity can be shown through a roundabout method using logarithms, while the second identity can be simplified using the natural logarithm. Both identities involve similar calculations and can be rewritten in different forms.
  • #1
quasar987
Science Advisor
Homework Helper
Gold Member
4,807
32
I don't get where these two identities come from:

[tex](logn)^{logn} = n^{log(logn)}[/tex]

and

[tex](logn)^{log(logn)} = e^{(log(logn))^2}[/tex]
 
Physics news on Phys.org
  • #2
I can only think of this roundabout way to show the first:

log(n)^log(n) = x
log(log(n)^log(n)) = log(x)
log(n) * log(log(n)) = log(x)
10^(log(n) * log(log(n))) = 10^log(x)
(10^log(n))^log(log(n)) = x
n^log(log(n)) = x

So log(n)^log(n) = n^log(log(n)).
 
  • #3
If you're using the natural logarithm, its usually better to use [itex]\ln[/itex] or [itex]\log_e[/itex] rather than [itex]\log[/itex] which can be interpreted in other ways (for example as [itex]\log_{10}[/itex] or as a log with unspecified base) depending on context.

The identities are similar:

[tex]n^{\ln(\ln(n))}=\left(e^{\ln(n)}\right)^{\ln(\ln(n))}=e^{\ln(n) \times \ln(\ln(n))}=e^{\ln(\ln(n)) \times \ln(n)}=\left(e^{\ln(\ln(n))}\right)^{\ln(n)}=\left(\ln(n)\right)^{\ln(n)}[/tex]

[tex]e^{\left(\ln(\ln(n))\right)^2}=e^{\ln(\ln(n)) \times \ln(\ln(n))}=\left(e ^{\ln(\ln(n))}\right)^{\ln(\ln(n))}=\left(\ln(n)\right)^{\ln(\ln(n))}[/tex]
 
  • #4
Oh, I see! Well thanks a bunch ! :smile:
 

1. What are logarithms and why are they important?

Logarithms are mathematical functions that represent the inverse of exponential functions. They are important because they allow us to simplify complex mathematical equations and make calculations more manageable.

2. What is the difference between natural logarithms and common logarithms?

Natural logarithms, represented by the symbol ln, use the base e (approximately 2.718) while common logarithms, represented by the symbol log, use the base 10. Both types of logarithms are used in different fields of mathematics and science.

3. How do you solve logn identities?

To solve logn identities, you can use properties of logarithms such as the product rule, quotient rule, and power rule. You can also use the inverse property to convert logarithmic equations into exponential form, which can make them easier to solve.

4. What are some real-world applications of logarithms?

Logarithms have many practical applications, such as in finance for calculating compound interest, in chemistry for measuring pH levels, and in computer science for measuring data storage and processing speeds.

5. How can understanding logn identities help in problem-solving?

Understanding logn identities can help in problem-solving by simplifying complex equations and making calculations more manageable. It also allows for the conversion of logarithmic equations into exponential form, which can make them easier to solve. Additionally, knowledge of logarithms can be applied to various real-world problems in different fields of study.

Similar threads

  • Programming and Computer Science
Replies
7
Views
1K
  • Programming and Computer Science
Replies
1
Views
1K
  • Engineering and Comp Sci Homework Help
Replies
1
Views
1K
  • Engineering and Comp Sci Homework Help
Replies
1
Views
1K
  • Engineering and Comp Sci Homework Help
Replies
5
Views
1K
Replies
3
Views
2K
  • Programming and Computer Science
Replies
2
Views
5K
  • Calculus and Beyond Homework Help
Replies
1
Views
4K
Replies
3
Views
1K
Back
Top