Logarithmic Properties and Common Logarithm

This page delves deeper into the properties of logarithms, introducing the common logarithm and presenting important algebraic properties of the natural logarithm.

The text begins by stating two crucial limits of the natural logarithm:

Highlight: • lim$x→+∞$ ln x = +∞ • lim$x→0+$ ln x = -∞

These limits describe the behavior of ln x as x approaches infinity and zero from the right, respectively.

The logarithme décimal (common logarithm, denoted as log) is then introduced:

Definition: The common logarithm is defined as log$x$ = ln$x$ / ln(10).

A graph comparing the natural and common logarithms is provided, illustrating their similar shapes but different scales. Key points for the common logarithm are noted:

• log 1 = 0 • log 10 = 1 • log 100 = 2

The text then presents important algebraic properties of the natural logarithm:

Highlight: For a > 0, b > 0, and n ∈ ℝ: • ln(a × b) = ln a + ln b • ln(a/b) = ln a - ln b • ln$a^n$ = n ln a • ln$1/a$ = -ln a • ln(√a) = ½ ln a

Vocabulary: "Propriétés algébriques" means algebraic properties, which are fundamental rules for manipulating logarithmic expressions.

The page concludes by noting that these properties also apply to the common logarithm (log x).

Finally, the derivative of a composite logarithmic function is presented:

Formula: For u$x$ > 0, if f$x$ = ln(u$x$), then f'$x$ = u'$x$ / u$x$.

This formula is crucial for differentiating more complex logarithmic expressions and is widely used in calculus.

Natural Logarithm Definition and Properties

The logarithme népérien (natural logarithm) is a crucial concept in mathematics, defined as the inverse of the exponential function. This page introduces its definition, key properties, and function behavior.

Definition: For a > 0, ln a is the unique real solution to the equation e^x = a. In other words, if e^x = a, then x = ln a.

Example: Two important cases are highlighted: • e^x = 1 ⇒ x = ln 1 = 0 • e^x = e ⇒ x = ln e = 1

The text also introduces a fundamental property of natural logarithms:

Highlight: For all x in $0, +∞$, ln$e^x$ = x, and for all real x, e^(ln x) = x.

The function f$x$ = ln x is then studied in detail:

1. It is defined and differentiable on the interval $0, +∞$.
2. Its derivative is f'$x$ = 1/x.
3. Since x > 0 and 1/x > 0, the derivative is always positive, making the function strictly increasing.

Vocabulary: The term "strictement croissante" means strictly increasing, indicating that as x increases, ln x also increases, but at a decreasing rate.

The page concludes with a graph illustrating the behavior of the natural logarithm function, showing its key points such as ln(1) = 0 and ln$e$ = 1.