Logarithm

The logarithm is defined by the formula $$\log(x) = \displaystyle\int_1^x \dfrac{1}{t} dt.$$

Properties

Proposition: $\displaystyle\int \log(z) dz = z \log(z)-z$

Proof: █

Theorem: For $|z|<1$, $$\log(1+z) = -\displaystyle\sum_{k=1}^{\infty} \dfrac{(-1)^k z^k}{k}.$$

Proof: █