3.7 The logarithmic function 85
0 Exercises 39
EXAMPLE 3.27What notto do.
A surprisingly common error is to put
ln 1 (x 1 + 1 y) 1 = 1 ln 1 x 1 + 1 ln 1 y
This is not in general true. For example,
ln 1 (1 1 + 1 2) 1 = 1 ln 1 3 but ln 111 + 1 ln 121 = 1 ln 1 2 (ln 111 = 1 0)
The only case for which ln 1 (x 1 + 1 y) 1 = 1 ln 1 x 1 + 1 ln 1 y is when x 1 + 1 y 1 = 1 xy; that is, when
x 1 = 1 y 2 (y 1 − 1 1).
Before the invention of the microchip and of the pocket calculator in the early 1970’s,
the ordinary logarithm was used mainly as an aid to long multiplication and division;
for example, the multiplication of numbers can be replaced by the addition of their
logarithms. There are now only a few uses of log
10in the physical sciences; for example
in the definitions of pH as a measure of hydrogen-ion concentration, and of pKwhere
Kis an equilibrium constant.
EXAMPLE 3.28pH as a measure of hydrogen-ion concentration
The pH of an aqueous solution is defined as
pH 1 = 1 −log
10[H
+]
where [H
+] is the ‘hydrogen-ion concentration’ in units of mol dm
− 3(moles per litre).
Then
[H
+] 1 = 110
−pHmol dm
− 3For example, a pH of 7 (neutral) corresponds to [H
+] 1 = 110
− 7mol dm
− 3.
ln( ) ln( ) ln( ) ln( ) ln( ) 11 111
21 2−+ + − −= −− +
−xx xxx−−−
=
−
+−
=
−
−
ln( )
ln
()
()()
ln
1
1
11
1
1
222x
x
xx
x
x
==ln1 0