# E^ln(x)=x Is it true always?

• vkash
In summary, the conversation discusses the function f(x)=eln(sin(x)) and whether it can be written as f(x)=x. The expert explains that this is not possible because the first function has no solution for negative numbers while the second one does. It is also mentioned that this is only true for positive real numbers and can be extended to negative numbers with the use of complex logarithms. The expert also suggests using x = -e^{\ln(-x)} or x = \frac{x}{|x|}e^{ln|x|} to avoid dealing with complex numbers.
vkash
consider this f(x)=eln(sin(x)) f:R-->R.
Can we write this function defination like this f(x)=x f:R-->R
I think no because if we put x as any negative number in first function(function in first line) then there will no solution exist for this but if we put x as any negative number in second function then their will be a solution.
So does it mean that eln(x)=x Is not true always.
ln represent natural log wih base e. R represent set of all real numbers.

It's only true for x greater than zero, so the positive real numbers onto the real numbers. In my limited experience I believe it is true for any log base.

If you extend the logarithms to negative numbers (to http://en.wikipedia.org/wiki/Complex_logarithm" ), you do have
$$e^{\ln x}=x$$

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S_Happens said:
It's only true for x greater than zero, so the positive real numbers onto the real numbers. In my limited experience I believe it is true for any log base.
thanks for confirming my thought.

dalcde said:
If you extend the logarithms to negative numbers (to http://en.wikipedia.org/wiki/Complex_logarithm" ), you do have
$$e^{\ln x}=x$$

I have not read any such thing.

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Then I presume you haven't dealt with functions of a complex variable.

For real numbers,
$$e^{ln(x)}= x$$
for any x such that ln(x) is defined- i.e. for positive real numbers.

To avoid getting into complex numbers, can't you just use $x = -e^{\ln(-x)}$ for $x<0$?

Or for any nonzero (real) x, $x = \frac{x}{|x|}e^{ln|x|}$?

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vkash said:
I have not read any such thing.
I apologize if I have gone too far, but my point is that if you somehow extend the logarithms to allow negative numbers, $e^{\ln x}=x$ is true for all x.

## 1. What is the equation E^ln(x)=x used for?

The equation E^ln(x)=x is used to simplify logarithmic expressions and to convert between exponential and logarithmic forms.

## 2. Does this equation hold true for all values of x?

Yes, this equation holds true for all positive values of x.

## 3. Is E^ln(x) equal to x for negative values of x?

No, this equation only holds true for positive values of x. For negative values of x, the equation E^ln(x)=x does not hold true.

## 4. Can this equation be used to solve for x in other equations?

Yes, this equation can be used to solve for x in certain exponential and logarithmic equations. However, it is important to check if the equation can be simplified before using this method.

## 5. Why is E^ln(x) equal to x?

This is because the natural logarithm (ln) and the natural exponential (e) are inverse functions of each other. This means that they "cancel out" when used together, resulting in the original value of x.

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