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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.

Робочий приклад: y = e^x

Algebra · Functions

Graph and analyse e^(x)

y = e^{x}

Крок за кроком

  1. e^{x}

    An expression in x. Here is what it does.

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

    Derivative (slope).

Показати відповідь
e^{x}

Symbols used here

f'(x),\ \frac{dy}{dx}
derivative
Instantaneous rate of change; slope of the graph.
e
Euler's number
2.71828…, the base whose exponential is its own derivative.
|x|
absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.
\binom{n}{k}
binomial coefficient, "n choose k"
Number of k-element subsets of n things: n!/(k!(n−k)!).
\pi
pi
Ratio of a circle's circumference to its diameter, 3.14159…
i
imaginary unit
i² = −1.
\theta
theta
The usual name for an angle.
\log_b x,\ \ln x
logarithm, natural log
The exponent b must be raised to for x; ln uses base e.
\mathbf{v},\ \vec{v}
vector
A quantity with magnitude and direction; a column of numbers.

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.

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