ALG-06

Exponential a^x

y = a^x = e^{x ln a} for a > 0.

Reading speed

ExponentialExponential

Governing equation

y=a^x

$x \in \left[-6, 8\right]$

where

a: Base (—)
x: Exponent (—)
y: a^x (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

y=a^x

$x \in \left[-6, 8\right]$

$x = -6$ $y = 0.016$

Along the curve

②

Domain of the variable — blue

x_{\min} = -6,\quad x_{\max} = 8

$x \in \left[-6, 8\right]$

$x = -6$ $x = -6$

Along the curve

③

Response curve — green

y=a^x

$x \in \left[-6, 8\right]$

$x = -6$ $y = 0.016$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}y}{\mathrm{d}x}

$x \in \left[-6, 8\right]$

$x = -5.912$ $y^{\prime} = 0.012$

Along the curve

Lecture brief

Historical brief

From al-Khwārizmī’s restoration to the quadratic formula, AM–GM, De Moivre and logarithms, algebra is the closed-form skeleton under every lab. These sheets solve, bound and compound. This sheet (ALG-06 — Exponential a^x) is the form associated with Exponential. Working symbols: $a$ , $x$ $\rightarrow$ $y$ . The exponential is the inverse of the logarithm. a^{x+y} = a^x a^y.

Purpose

Purpose: compute

y

from

a

x

in Algebra via

y=a^x

y = a^x = e^{x ln a} for a > 0. Use it when a real algebra question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

a = 2.000\,\mathrm{—}

x = 3.000\,\mathrm{—}

, the governing relation

y=a^x

yields

y = 8.0000\,\mathrm{—}

. Positive base, real exponent. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Base a2 —Exponent x3 —

Outputs

a^x y8.0000 —

Reading speed

Watch on YouTube

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Narration of this film

Positive base, real exponent.

The exponential is the inverse of the logarithm. a^{x+y} = a^x a^y.

Reading speed

Watch on YouTube