ALG-05

Change-of-base logarithm

log_b a = ln a / ln b.

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LogarithmsChange of base

Governing equation

\log_b a=\ln a/\ln b

where

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

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

Watch on YouTube

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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