Lecture

Exponential Functions and Logarithms

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Description

This lecture covers the definitions and graphs of exponential and logarithmic functions, including the properties of the exponential function, such as being strictly increasing and unbounded. It also explores fundamental exponential relations, the exponential function with base e, and the logarithmic function with different bases. The lecture delves into the graphs of exponential functions with various bases, highlighting their continuity, differentiability, and monotonicity. Additionally, it discusses the logarithmic function with different bases, emphasizing their bijectivity and the concept of inverse functions. The most commonly used logarithms, including natural logarithm, base 10 logarithm, and base 2 logarithm, are also presented.

In MOOC

Trigonometric, Logarithmic and Exponential Functions

This course provides the fundamental knowledge related to these functions. It is structured around 9 chapters spread over 12 weeks: The angle and its measurement Trigonometric functions sine and cosine and their inverses Trigonometric functions tangent and cotangent and their inverses Trigonometric formulas (addition, bisection, etc.) Harmonic oscillations Trigonometric equations Triangle relations (sine and cosine theorems) Continuity and derivability of trigonometric functions Logarithmic and exponential

Official source

Ontological neighbourhood

Mathematics

Analysis: Complex analysis

Discrete mathematics: Graph theory

Related concepts (32)

Exponential function

The exponential function is a mathematical function denoted by or (where the argument x is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the notion of exponentiation (repeated multiplication), but modern definitions (there are several equivalent characterizations) allow it to be rigorously extended to all real arguments, including irrational numbers.

Logarithmic scale

A logarithmic scale (or log scale) is a way of displaying numerical data over a very wide range of values in a compact way. As opposed to a linear number line in which every unit of distance corresponds to adding by the same amount, on a logarithmic scale, every unit of length corresponds to multiplying the previous value by the same amount. Hence, such a scale is nonlinear: the numbers 1, 2, 3, 4, 5, and so on, are not equally spaced. Rather, the numbers 10, 100, 1000, 10000, and 100000 would be equally spaced.

Inverse trigonometric functions

In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.

Logarithm

In mathematics, the logarithm is the inverse function to exponentiation. That means that the logarithm of a number x to the base b is the exponent to which b must be raised to produce x. For example, since 1000 = 103, the logarithm base 10 of 1000 is 3, or log10 (1000) = 3. The logarithm of x to base b is denoted as logb (x), or without parentheses, logb x, or even without the explicit base, log x, when no confusion is possible, or when the base does not matter such as in big O notation.

Chordal graph

In the mathematical area of graph theory, a chordal graph is one in which all cycles of four or more vertices have a chord, which is an edge that is not part of the cycle but connects two vertices of the cycle. Equivalently, every induced cycle in the graph should have exactly three vertices. The chordal graphs may also be characterized as the graphs that have perfect elimination orderings, as the graphs in which each minimal separator is a clique, and as the intersection graphs of subtrees of a tree.

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