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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.

Worked example · y = e^x

Algebra · Functions

Graph and analyse e^(x)

y = e^{x}

Step by step

  1. e^{x}

    An expression in x. Here is what it does.

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

    Derivative (slope).

Reveal the answer
e^{x}

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Symbols used here

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How to: Exponential and logarithmic functions

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

Questions people ask

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.

More in Precalculus

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