Question

What is the natural log of -0.2877?

Answer 1

If you are talking about `ln` as a Real valued function of Real numbers then `ln(-0.2877)` is undefined.

The principal Complex natural logarithm is:

`ln(-0.2877) = ln(0.2877)+pii ~~ -1.2458+pii`

Explanation:

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Real natural logarithm

The function `e^x` with domain `(-oo, oo)` and range `(0, oo)` is one to one. So it has a well defined inverse function `ln(x)` with domain `(0, oo)` and range `(-oo, oo)`

Since negative numbers (and zero) are not in the range of `e^x`, they are not in the domain of the inverse Real logarithm function `ln(x)`. So `ln(-0.2877)` is undefined.

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Complex natural logarithm

The function `e^z` has domain `CC` and range `CC "\" { 0 }`

It is many to one, e.g. `e^0 = 1 = e^(2pii)`, so it does not have a well defined inverse function. However, if we restrict the domain to `{ z in CC : -pi < Im(z) <= pi }` then it has a well defined inverse:

`ln(z) = ln abs(z) + Arg(z)i`

In particular, for negative Real numbers, we have:

`ln(x) = ln(-x) + pii`

Hence:

`ln(-0.2877) = ln(0.2877)+pii ~~ -1.2458+pii`