Question

Am I right in thinking that `ln(-6)` is undefined ?

Answer 1

`"correct"`

Explanation:

`"the log function is not defined for negative values"`

`"the graph of ln should confirm this for you"`
graph{lnx [-10, 10, -5, 5]}

Answer 2

It depends...

Explanation:

As a real valued function of real numbers the function `e^x` is a one to one function from `(-oo, oo)` onto `(0, oo)`.

Therefore it has a well defined inverse function `ln(x)` from `(0, oo)` onto `(-oo, oo)`.

This real valued logarithm is not defined for any real number in `(-oo, 0]`

However, `e^x` can also be considered as a complex valued function of complex numbers.

As such, it is a many to one function from `CC` onto `CC"\"{0}`.

Since it is many to one, the complex valued logarithm `ln x` that extends the real valued logarithm has multiple branches. The principal logarithm is usually taken to have values in:

`{ x+yi : x in RR, y in (-pi, pi] }`

If `x` is any negative real number, then the principal value of its complex logarithm is:

`ln x = ln (-x) + pi i`

So, for example:

`ln(-6) = ln(6) + pi i`

In general:

`ln(r(cos theta + i sin theta)) = ln r + theta i" "` for `theta in (-pi, pi]`

Answer 3

Yes

Explanation:

Say you have the following expression:

`ln(-6) = x`

This directly translates, by the definition of the natural log, to

`e^x = -6`

`e` is a positive number equal to approximately 2.72. No matter how many times you multiply a positive number by itself, you will always still get a positive number, so `e^x` can never equal `-6`, and the answer is considered undefined.