The Chemistry Maths Book, Second Edition

3.7 The logarithmic function 83

3.7 The logarithmic function

The logarithmic function

6

is the inverse function of the exponential:

if y 1 = 1 a

x

then x 1 = 1 log

a

1 y (3.34)

and log

a 1

yis called the logarithm to base aof y. The most important logarithmic

functions are the ordinary logarithm, to base 10,

y 1 = 110

x

, x 1 = 1 log

10

1 y 1 = 1 lg y (3.35)

and the natural logarithm(sometimes called the Napierian logarithm), to base e,

y 1 = 1 e

x

, x 1 = 1 log

e

1 y 1 = 1 ln 1 y (3.36)

The ordinary logarithm log

10

is sometimes given the symbol lg. The natural logarithm

log

e

is nearly always given the symbol ln.

It follows from equations (3.34) to (3.36) that

y 1 = 1 log

a

a

y

1 = 1 log

10

10

y

1 = 1 ln 1 e

y

(3.37)

EXAMPLE 3.23

log

2

2

3

1 = 1 3, log

10

10

3

1 = 1 3, lg 110

− 2

1 = 1 −2,

log

e

e

3

1 = 1 3, ln 1 e

− 122

1 = 1 − 12 2, log

a

a

0

1 = 1 log

a

11 = 10

0 Exercises 37

We note that the logarithm of 1 to any base is zero.

The graph ofln 1 xand of its inverse function e

x

are shown in Figure 3.19.

7

6

John Napier (1550–1617), Scottish baron and amateur mathematician, published his invention of what

he called logarithms in the Mirifici logarithmorum canonis descriptio(A description of the wonderful canon

of logarithms) in 1614. Napier’s logarithms were based on a logarithm of 10

7

1 = 10. The first table of common

logarithms, withlog 111 = 10 andlog 1101 = 11 , was published, after a famous consultation with Napier, by Henry

Briggs (1561–1630), professor of geometry at Oxford, in the Arithmetica logarithmicain 1624. Logarithms greatly

simplified computations involving multiplication and division.

7

The graph of a log function was first drawn in 1646 by Evangelista Torricelli (1608–1647).

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Figure 3.19