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Logarithm

Definition​

Definition

The natural logarithm, noted ln⁡\ln, is the unique primitive on R+∗\mathbb R_+^* of the function x→1xx \to \frac{1}{x} that vanishes at 1. It can be also written

∀x∈R+∗ln⁡x=∫1xduu.\forall x \in \mathbb R_+^* \quad \ln x = \int_1^x\frac{du}{u}.

Results​

Proposition

The natural logarithm verify

∀x∈R+∗∀y∈R+∗ln⁡(xy)=ln⁡(x)+ln⁡(y).\forall x \in \mathbb R_+^* \quad \forall y \in \mathbb R_+^* \quad \ln(xy) = \ln(x) + \ln(y).
Corollary
  1. We have : ∀x∈R+∗∀y∈R+∗ln⁡(xy)=ln⁡(x)−ln⁡(y)\forall x \in \mathbb R_+^* \quad \forall y \in \mathbb R_+^* \quad \ln\left (\frac{x}{y} \right ) = \ln(x) - \ln(y)
  2. We have : ∀x∈R+∗∀n∈Zln⁡(xn)=nln⁡x.\forall x \in \mathbb R_+^* \quad \forall n \in \mathbb Z \quad \ln(x^n) = n\ln x.
Proposition

The natural logarithm is a strictly increasing bijection of ]0,+∞[]0, +\infty[ on R\mathbb R verifying

lim⁡x→+∞ln⁡x=+∞andlim⁡x→0ln⁡x=−∞.\lim_{x \to +\infty} \ln x = + \infty \qquad \text{and} \qquad \lim_{x \to 0}\ln x = -\infty.