LOGARITHM RULE COPIED!

Logarithm Product Rule

logb(M · N) = logbM + logbN
Concept: Multiplication inside becomes Addition outside.

Logarithm Key Concepts 🪵

b
The Base

The number that is being raised to a power. Common bases are 10 and 'e'.

x
The Argument

The value we are taking the log of. It's the result of the original exponent.

y
The Exponent

The power to which the base must be raised to produce the argument x.

ln
Natural Log

A special log with base 'e' (approx 2.718). Used heavily in calculus.

log
Common Log

When no base is written, it is assumed to be base 10.

10x
Antilogarithm

The inverse operation that 'undoes' a logarithm to find the original value.

+
Product → Sum

Multiplying numbers inside a log is the same as adding their logs.

−
Ratio → Diff

Dividing numbers inside a log is the same as subtracting their logs.

x > 0
The Domain

You cannot take the log of zero or a negative number. It must be positive!

Legacy of the Logarithm 📜

log

1614 AD | JOHN NAPIER

Scottish mathematician John Napier published the first table of logarithms. His goal was to simplify the huge calculations needed for astronomy and navigation.

1617 AD | HENRY BRIGGS

Henry Briggs visited Napier and suggested using 'Base 10'. This made logarithms much more practical for everyday science and created the "Common Log" we use today.

1620s | ANALOG COMPUTING

Logarithms led to the invention of the Slide Rule. Before digital calculators, engineers used this "analog computer" based on logs to build bridges and even go to the Moon!

"Did you know? Before calculators, mathematicians spent years hand-writing 'Log Tables' so scientists could solve complex problems in minutes instead of weeks!"