Beautiful Math

How this idea was born

Additive change is comfortable: walk three more miles. Multiplicative change is different: double the grain each season, halve the drug concentration each half-life, compound interest each year. When quantities grow or decay by repeated factors, linear intuition lies.

Astronomers in Napier's day drowned in multiplications for sine tables and planetary models. In 1614 John Napier published logarithms so that hard products could become easier sums. MacTutor is careful: Napier's original logarithms are not the modern

\log_b

you meet in school (in particular, Nap.log 1 is not 0). Henry Briggs traveled to Edinburgh; their discussions produced the practical program of base-10 logarithms with

\log 1 = 0

. Briggs gives Napier credit for key framing even when Briggs built the tables.

For a modern rereader,

a^x

(with

a > 0

a \neq 1

) is the growth machine, and

\log_a

is its inverse: it answers "to what power must I raise

a

to get this?"

First principles

For base

a > 0

a \neq 1

y = a^x \quad \Longleftrightarrow \quad x = \log_a y

with

y > 0

. Domain of

\log_a

(0, \infty)

; range is all reals.

Laws (the reason logs help):

\log_a(xy) = \log_a x + \log_a y, \quad \log_a(x/y) = \log_a x - \log_a y, \quad \log_a(x^k) = k\log_a x

Products become sums. That is Napier's computational dream in modern clothes.

Common bases:

\log_{10}

(common),

\ln = \log_e

(natural),

\log_2

(information, CS). Change of base:

\log_b x = \frac{\log_k x}{\log_k b}

Growth versus decay: if

a > 1

a^x

grows as

x

increases. If

0 < a < 1

, it decays toward 0. Graphs: exponential curves, logarithmic slow climbs.

Worked intuition

Inverse as undo

2^5 = 32

, then

\log_2 32 = 5

. The log recovers the exponent. Composition:

\log_a(a^x) = x

and

a^{\log_a y} = y

(on the proper domains).

Compound interest sketch

A = P(1+r)^t

is exponential in

t

. Solving for time uses a log:

t = \log(A/P)/\log(1+r)

Half-life

If quantity halves every

T

years,

N(t) = N_0 (1/2)^{t/T}

. Taking

\ln

isolates

t

when you know a target

N

Graph literacy

y = 2^x

passes through

(0,1)

, rises;

y = \log_2 x

passes through

(1,0)

, defined only for

x > 0

. They are reflections across

y = x

, the geometric signature of inverse functions from the functions lesson.

Common confusions

Writing $\log(x+y) = \log x + \log y$ . False. The sum law needs a product inside.

Allowing $\log$ of zero or negatives (in reals). Domain is positive reals only.

Equating Napierian logs with $\ln$ without comment. Related historical family; not identical objects. Say "Napier's logarithms" when you mean 1614.

Thinking $a^{x+y} = a^x + a^y$ . Exponents add inside a product of powers:

a^{x+y} = a^x a^y

History and stories

Napier's 1614 Mirifici Logarithmorum Canonis Descriptio aimed at calculation. Briggs's visits (1615-1616) and subsequent Arithmetica Logarithmica made common logs a working tool for navigators and astronomers. Slide rules later embodied the same additive trick mechanically.

The constant

e

and natural logarithms grow from later calculus and series; do not back-date

\ln

into Napier's first tables. The ethic travels: when multiplication is the bottleneck, change coordinates so addition does the work.

Real world

Finance: compound interest and continuous compounding limits.
Science: radioactive decay, cooling models, Richter and pH (log scales).
Computing: bit depth and information as $\log_2$ .
Sound: decibels as logarithmic ratios of intensity.

Open when you want the machine-learning connection. Skip freely.

Field notes, extra examples, and glossary live here when you want more than the core path.

Carry this forward

Exponentials multiply; logs invert and turn products into sums.
Napier/Briggs: computation as compassion, with honest historical nuance.
Next: sequences and fitting, where patterns meet data.

Exponentials and logarithms