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Exponential and logarithmic functions

Growth and decay, the number e, log rules, and change of base.

bˣ grows by the same factor for each unit of x; its inverse log_b x asks which exponent produces x. The rules log(ab) = log a + log b and log aᵏ = k log a are the exponent rules read backwards. Picture it: the exponential curve and its log are mirror images across y = x. Think it: e is the base whose growth rate equals its value, which is why calculus prefers it.

작업 예제 · y = e^x

Algebra · Functions

Graph and analyse e^(x)

y = e^{x}

단계별로

  1. e^{x}

    An expression in x. Here is what it does.

  2. \frac{d}{dx} = e^{x}

    Derivative (slope).

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e^{x}

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어떻게: Exponential and logarithmic functions

  1. An expression in x. Here is what it does.
  2. Derivative (slope).

사람들이 묻는 질문

What is a function, really?

A rule that assigns exactly one output to each input. The vertical-line test on a graph is the same idea: no input may have two outputs.

Why do we need complex numbers?

Because x² + 1 = 0 has no real solution, and allowing one new number i with i² = −1 makes every polynomial equation solvable. They then turn out to describe rotation, waves and alternating current more naturally than real numbers do.

에 더 Precalculus

Complex numbersPolynomial functionsRational functionsSequences and seriesThe binomial theoremConic sectionsVectorsPolynomial division and the remainder theoremParametric equations and polar coordinates