# What is the natural log of -0.2877?

Jun 26, 2016

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

The principal Complex natural logarithm is:

$\ln \left(- 0.2877\right) = \ln \left(0.2877\right) + \pi i \approx - 1.2458 + \pi i$

#### Explanation:

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Real natural logarithm

The function ${e}^{x}$ with domain $\left(- \infty , \infty\right)$ and range $\left(0 , \infty\right)$ is one to one. So it has a well defined inverse function $\ln \left(x\right)$ with domain $\left(0 , \infty\right)$ and range $\left(- \infty , \infty\right)$

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 $\ln \left(x\right)$. So $\ln \left(- 0.2877\right)$ is undefined.

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Complex natural logarithm

The function ${e}^{z}$ has domain $\mathbb{C}$ and range $\mathbb{C} \text{\} \left\{0\right\}$

It is many to one, e.g. ${e}^{0} = 1 = {e}^{2 \pi i}$, so it does not have a well defined inverse function. However, if we restrict the domain to $\left\{z \in \mathbb{C} : - \pi < I m \left(z\right) \le \pi\right\}$ then it has a well defined inverse:

$\ln \left(z\right) = \ln \left\mid z \right\mid + A r g \left(z\right) i$

In particular, for negative Real numbers, we have:

$\ln \left(x\right) = \ln \left(- x\right) + \pi i$

Hence:

$\ln \left(- 0.2877\right) = \ln \left(0.2877\right) + \pi i \approx - 1.2458 + \pi i$