Inverse of the natural logarithms

In summary, the conversation discusses a problem with finding the inverse of a function and determining its range and domain. The solution includes showing that the function is strictly increasing and identifying the inverse's domain and range. The participants also welcome a new member to the forum and provide helpful tips for solving the problem.
  • #1
Wingman
8
0
Hi Guys, i am new to this forums and my english is poor, but i will do my best.

I got stuck with this problem, i think it's quite easy, but i get the wrong answer :frown:

F(x) = ln(1+e^x)

1. Show that it has an inverse
2. What is the Range And the Domain of the inverse.

I really appreciate a good solution so i can learn from my mistakes.
 
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  • #2
1. Show that F is a strictly increasing function.
2. The domain of the inverse is the range of F and vice versa.
3. Welcome to PF!
 
  • #3
so the inverse must be ln(e^x-1)?
 
  • #4
Yeah, that seems right.
 
  • #5
The domain must be then; ]0, infinity[ And the range]-1, Infinity[ or am i wrong?
 
  • #6
Your domain is correct, but the range of the inverse is from negative to positive infinity.
 
  • #7
of course, how stuiped of me :blushing: Thx for the help! :biggrin:
 

1. What is the inverse of the natural logarithm function?

The inverse of the natural logarithm function is the exponential function, denoted as ex. It is the function that undoes the effect of the natural logarithm function, meaning that for any given value of x, ex will give the result of the natural logarithm function applied to that value.

2. How is the inverse of the natural logarithm function defined?

The inverse of the natural logarithm function is defined as ex = y if and only if ln(y) = x, where e is the mathematical constant approximately equal to 2.71828 and ln is the natural logarithm function.

3. What is the graph of the inverse of the natural logarithm function?

The graph of the inverse of the natural logarithm function is a curve that passes through the points (0,1) and (1, e). It is an increasing function that approaches infinity as x approaches infinity and approaches 0 as x approaches negative infinity. The graph is also symmetrical about the line x = y, meaning that it is its own reflection when rotated 90 degrees clockwise.

4. How is the inverse of the natural logarithm function used in real life?

The inverse of the natural logarithm function is used in a variety of fields, including finance, biology, and physics. In finance, it is used to calculate compound interest and in biology, it is used to model population growth. In physics, it is used to describe phenomena such as radioactive decay and electrical circuits.

5. Can the inverse of the natural logarithm function be simplified?

Yes, the inverse of the natural logarithm function can be simplified using logarithmic rules. For example, eln(x) = x, and ln(ex) = x. These rules can be used to simplify more complex equations involving the inverse of the natural logarithm function.

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