ALG-26

Change of base

Any base.

Reading speed

Algebralog change

Governing equation

\\log_b a=\\ln a/\\ln b

$a \in \left[0.1, 100\right]$

where

a: a (—)
b: b (—)
\log_b a: log (—)

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

\\log_b a=\\ln a/\\ln b

$a \in \left[0.1, 100\right]$

$a = 0.1$ $\log_b a = -3.322$

Along the curve

②

Domain of the variable — blue

a_{\min} = 0.1,\quad a_{\max} = 100

$a \in \left[0.1, 100\right]$

$a = 0.1$ $a = 0.1$

Along the curve

③

Response curve — green

\\log_b a=\\ln a/\\ln b

$a \in \left[0.1, 100\right]$

$a = 0.1$ $\log_b a = -3.322$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\log_b a}{\mathrm{d}a}

$a \in \left[0.1, 100\right]$

$a = 0.728$ $\log_b a^{\prime} = 2.994$

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-26 — Change of base) is the form associated with log change. Working symbols: $a$ , $b$ $\rightarrow$ $\log_b a$ . Any base.

Purpose

Purpose: compute

\log_b a

from

a

b

in Algebra via

\\log_b a=\\ln a/\\ln b

Any base. 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 = 8.000\,\mathrm{—}

b = 2.000\,\mathrm{—}

, the governing relation

\\log_b a=\\ln a/\\ln b

yields

\log_b a = 3.0000\,\mathrm{—}

. Any base. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

a a8 —b b2 —

Outputs

log \log_b a3.0000 —

Reading speed

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Narration of this film

Any base.

Reading speed

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