ALG-21

Logarithm quotient rule

ln(x/y) = ln x − ln y.

Reading speed

LogarithmsLog quotient

Governing equation

\ln(x/y)=\ln x-\ln y

$x \in \left[0.01, 80\right]$

where

x: x (—)
y: y (—)
\ln(x/y): ln of quotient (—)
\ln x-\ln y: Difference of lns (—)

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

\ln(x/y)=\ln x-\ln y

$x \in \left[0.01, 80\right]$

$x = 0.01$ $\ln(x/y) = -5.704$

Along the curve

②

Domain of the variable — blue

x_{\min} = 0.01,\quad x_{\max} = 80

$x \in \left[0.01, 80\right]$

$x = 0.01$ $x = 0.01$

Along the curve

③

Response curve — green

\ln(x/y)=\ln x-\ln y

$x \in \left[0.01, 80\right]$

$x = 0.01$ $\ln(x/y) = -5.704$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\ln(x/y)}{\mathrm{d}x}

$x \in \left[0.01, 80\right]$

$x = 0.513$ $\ln(x/y)^{\prime} = 4.593$

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-21 — Logarithm quotient rule) is the form associated with Log quotient. Working symbols: $x$ , $y$ $\rightarrow$ $\ln(x/y)$ , $\ln x-\ln y$ . The logarithm converts division into subtraction — the inverse of the exponential homomorphism.

Purpose

Purpose: compute

\ln(x/y)

\ln x-\ln y

from

x

y

in Algebra via

\ln(x/y)=\ln x-\ln y

ln(x/y) = ln x − ln y. Use it when a real algebra question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

x = 12.000\,\mathrm{—}

y = 3.000\,\mathrm{—}

, the governing relation

\ln(x/y)=\ln x-\ln y

yields

\ln(x/y) = 1.3863\,\mathrm{—}

\ln x-\ln y = 1.3863\,\mathrm{—}

. Two positive numbers. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

x x12 —y y3 —

Outputs

ln of quotient \ln(x/y)1.3863 —
Difference of lns \ln x-\ln y1.3863 —

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Narration of this film

Two positive numbers.

The logarithm converts division into subtraction — the inverse of the exponential homomorphism.

Reading speed

Watch on YouTube