Natural Logarithm Function and Properties

The page introduces the fundamental concepts and properties of the natural logarithm function, known in French as fonction logarithme népérien.

Definition: The natural logarithm of a strictly positive real number a is defined as the unique solution to the equation eˣ = a, denoted as ln$a$.

Vocabulary: The domain of the natural logarithm function is $0,+∞$, mapping to all real numbers.

Highlight: The exponential and natural logarithm functions are inverse functions of each other, with their graphs being symmetric about the line y = x.

Key Properties:

• The function is differentiable on $0,+∞$ with derivative 1/x
• It is strictly increasing and concave on its domain
• Essential limits include lim(x→0⁺) ln$x$ = -∞ and lim$x→+∞$ ln$x$ = +∞

Example: Important identities include:

• ln(xy) = ln$x$ + ln$y$
• ln(x/y) = ln$x$ - ln$y$
• ln(√x) = ½ln$x$
• ln(xⁿ) = nln$x$

Highlight: The function's behavior at key points includes:

• ln(1) = 0
• ln$e$ = 1
• For any x, y > 0: ln$x$ < ln$y$ if and only if x < y

The page concludes with important limit properties and composition rules, essential for advanced calculus applications.