ALG-15

Logarithm product rule

log(xy) = log x + log y. The sheet checks both sides in ln.

Reading speed

LogarithmsLog product

Governing equation

\ln(xy)=\ln x+\ln y

$x \in \left[0.01, 50\right]$

where

x: x (—)
y: y (—)
\ln(xy): ln of product (—)
\ln x+\ln y: Sum 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.

00:0 / 00:08

①

Governing law — red

\ln(xy)=\ln x+\ln y

$x \in \left[0.01, 50\right]$

$x = 0.01$ $\ln(xy) = -2.996$

Along the curve

②

Domain of the variable — blue

x_{\min} = 0.01,\quad x_{\max} = 50

$x \in \left[0.01, 50\right]$

$x = 0.01$ $x = 0.01$

Along the curve

③

Response curve — green

\ln(xy)=\ln x+\ln y

$x \in \left[0.01, 50\right]$

$x = 0.01$ $\ln(xy) = -2.996$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\ln(xy)}{\mathrm{d}x}

$x \in \left[0.01, 50\right]$

$x = 0.324$ $\ln(xy)^{\prime} = 6.611$

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-15 — Logarithm product rule) is the form associated with Log product. Working symbols: $x$ , $y$ $\rightarrow$ $\ln(xy)$ , $\ln x+\ln y$ . Because exp is a homomorphism: e^{a+b} = e^a e^b, so ln of a product is a sum.

Purpose

Purpose: compute

\ln(xy)

\ln x+\ln y

from

x

y

in Algebra via

\ln(xy)=\ln x+\ln y

log(xy) = log x + log y. The sheet checks both sides in ln. 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 = 2.000\,\mathrm{—}

y = 5.000\,\mathrm{—}

, the governing relation

\ln(xy)=\ln x+\ln y

yields

\ln(xy) = 2.3026\,\mathrm{—}

\ln x+\ln y = 2.3026\,\mathrm{—}

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

Calculator

Inputs

x x2 —y y5 —

Outputs

ln of product \ln(xy)2.3026 —
Sum of lns \ln x+\ln y2.3026 —

Reading speed

Watch on YouTube

Free library

Full library

Free PDF / open book

YouTube channels

00:0 / 00:08

Narration of this film

Two positive factors.

Because exp is a homomorphism: e^{a+b} = e^a e^b, so ln of a product is a sum.

Reading speed

Watch on YouTube