The Chemistry Maths Book, Second Edition

86 Chapter 3Transcendental functions

The pH is an example of the use of the logarithm as a scale of measure. Other

examples of logarithmic scales are the Richter scale for the ‘strength of earthquake’,

the bel scale of loudness, and the scale of star magnitudes.

0 Exercises 40–42

Conversion factors

The logarithm occurs in a number of expressions for physical properties and

processes, and the ordinary logarithm is still sometimes found, instead of the natural

logarithm, in the scientific literature and in textbooks. In general, the conversion

factor from one base, a, to another, b, is given by

log

b

x 1 = 1 log

b

a 1 × 1 log

a

x (3.42)

Thus, if x 1 = 1 a

y

, then

log

a

x 1 = 1 y

log

b

x 1 = 1 log

b

a

y

1 = 1 y 1 log

b

a 1 = 1 (log

a

x) 1 × 1 (log

b

a)

The conversion factors between ordinary and natural logarithms are therefore

lg 1 x 1 = 1 log

10

x 1 = 1 log

10

e 1 × 1 log

e

x 1 ≈ 1 0.43429448 1 × 1 log

e

x

ln 1 x 1 = 1 log

e

x 1 = 1 log

e

101 × 1 log

10

x 1 ≈ 1 2.30258093 1 × 1 log

10

x

3.8 Values of exponential and logarithmic functions

Table 3.3 shows values ofln 1 x, x 1 ln 1 x, e

x

, ande

−x

for a wide range of values of x.

Table 3.3

x ln 1 xx 1 ln 1 xe

x

e

−x

0–∞ 01 1

slowly

10

− 6

−13.8 −0.00001 1.000001 0.9999990

10

− 3

−6.9 −0.007 1.001 0.9990

1 0 0 2.7 0.37

10 2.3 23 21 × 110

4

51 × 110

− 5

10

2

4.6 460 31 × 110

43

41 × 110

− 44

10

3

6.9 6908

slowly fast fast

∞∞ ∞ ∞ 0