ALG-05

Change-of-base logarithm

log_b a = ln a / ln b.

Reading speed

LogarithmsChange of base

Governing equation

\log_b a=\ln a/\ln b

$a \in \left[0.01, 1000\right]$

where

a: Argument (—)
b: Base (—)
\log_b a: Logarithm (—)

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

$a \in \left[0.01, 1000\right]$

$a = 0.01$ $\log_b a = -6.644$

Along the curve

②

Domain of the variable — blue

a_{\min} = 0.01,\quad a_{\max} = 1000

$a \in \left[0.01, 1000\right]$

$a = 0.01$ $a = 0.01$

Along the curve

③

Response curve — green

\log_b a=\ln a/\ln b

$a \in \left[0.01, 1000\right]$

$a = 0.01$ $\log_b a = -6.644$

Along the curve

④

Rate of change — amber

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

$a \in \left[0.01, 1000\right]$

$a = 6.299$ $\log_b a^{\prime} = 0.819$

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-05 — Change-of-base logarithm) is the form associated with Change of base. Working symbols: $a$ , $b$ $\rightarrow$ $\log_b a$ . Any logarithm is a ratio of natural logs. The formula converts calculator ln into an arbitrary base.

Purpose

Purpose: compute

\log_b a

from

a

b

in Algebra via

\log_b a=\ln a/\ln b

log_b a = ln a / ln b. 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{—}

. a > 0, b > 0, b ≠ 1. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Argument a8 —Base b2 —

Outputs

Logarithm \log_b a3.0000 —

Reading speed

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Narration of this film

a > 0, b > 0, b ≠ 1.

Any logarithm is a ratio of natural logs. The formula converts calculator ln into an arbitrary base.

Reading speed

Watch on YouTube