Proof of limit derivation of Natural Logrithm

In summary, the natural logarithm can be expressed as the limit of n(x^(1/n)-1) as n approaches infinity. The speaker is requesting help in deriving this expression and has attempted to use the limit definition of e^x and the inverse function, but has only been able to obtain ((n*x)-1)^1/n. They also mention that they are on mobile and unable to easily display mathematical expressions.
  • #1
cmcraes
99
6
The natural logarithm can be expressed as
lim n(x^(1/n)−1)
n->ifinity

Can someone please derive this for me? I've tried using the limit definition of e^x and then applying f^-1(x) but i can only get
((n*x)-1)^1/n

any help is greatly appreciated thanks!
 
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  • #2
hi cmcraes! :smile:

(try using the X2 button just above the Reply box :wink:)

x = elogx ? :smile:​
 
  • #3
No that doesn't use limits nor does it prove anything other than the fact that the natural log and the exponential function cancel each other out

ps: I am on mobile so its not easy to do math print
 
  • #4
expand elogx/n :wink:
 

FAQ: Proof of limit derivation of Natural Logrithm

What is the proof of the limit derivation of the natural logarithm?

The proof of the limit derivation of the natural logarithm involves using L'Hopital's rule and the definition of the natural logarithm to show that the limit of ln(x) as x approaches 1 is equal to 1.

Why is the proof of the limit derivation of the natural logarithm important?

This proof is important in calculus and other areas of mathematics because it provides a way to find the derivatives of functions involving the natural logarithm, such as ln(x) and e^x.

What is L'Hopital's rule and how is it used in the proof?

L'Hopital's rule is a method for evaluating the limit of a ratio of two functions. In the proof of the limit derivation of the natural logarithm, it is used to simplify the limit of ln(x) as x approaches 1.

Can you explain the concept of the natural logarithm and its relationship to e?

The natural logarithm, denoted as ln(x), is the inverse function of the exponential function e^x. It is defined as the integral of 1/x from 1 to x. This relationship between the natural logarithm and e is what allows us to use L'Hopital's rule in the proof of the limit derivation.

How can the proof of the limit derivation of the natural logarithm be applied in real-world situations?

This proof has many applications in science and engineering, such as in calculating compound interest, population growth, and the decay of radioactive elements. It is also used in statistics and probability to model data that follows an exponential growth or decay pattern.

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