The logarithm is generally defined to be an inverse function for the exponential. If $x > 0$ is a real number and $b > 0$, $b \ne 1$, then the base-$b$ logarithm is defined by

$$\log_b(x) = y \iff b^y = x$$

The most commonly used bases are base $10$ and $2$ (which frequently arises in computer science), and particularly base $e$. The natural logarithm $\ln$ is defined to be $\log_e$.

Alternatively, the natural logarithm can be defined to be a primitive of the function $$f(t) = \frac{1}{t}$$ subject to the condition that $\ln{1} = 0$.

In the study of complex numbers, the solutions $a$ of $e^{a} = z$ are called complex logarithms. This uniquely specifies the modulus of $a$, but not its argument; as such, we define the principal logarithm $\operatorname{Log}(re^{i\theta}) = \ln{r} + i \theta$, with the restriction $-\pi < \theta \le \pi$ (or alternatively, $0 \le \theta < 2\pi$). This leads to a branch cut, or discontinuity - alternatively, the complex logarithm can be viewed as a multi-valued function.

Reference: Logarithm.

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