The Chemistry Maths Book, Second Edition

84 Chapter 3Transcendental functions

The graphs of all the logarithmic functions are similar, with properties

log 111 = 1 0, log 1 x 1 → 1 +∞ as x 1 → 1 ∞, log 1 x 1 → 1 −∞ as x → 0 (3.38)

We note that (real) log 1 xis not defined for negative values of x.

The combination properties of logarithms are the properties of the indices

(exponents) of the corresponding exponentials; for the natural logarithm:

ln 1 x 1 + 1 ln 1 y 1 = 1 ln 1 xy (3.39)

(3.40)

ln 1 x

n

1 = 1 n 1 ln 1 x (3.41)

EXAMPLE 3.24Addition of logarithms. To prove equation (3.39),

let x 1 = 1 e

a

and y 1 = 1 e

b

,so thatxy 1 = 1 e

a+b

Then, by definition (3.36)

ln 1 x 1 = 1 a,ln 1 y 1 = 1 b,ln 1 xy 1 = 1 a 1 + 1 b

Therefore ln 1 x 1 + 1 ln 1 y 1 = 1 ln 1 xy.

EXAMPLE 3.25Combinations of logarithms.

0 Exercises 38

EXAMPLE 3.26Simplify the expression ln 1 (1 1 − 1 x

2

) 1 + 1 ln 1 (1 1 + 1 x)

− 1

1 − 1 ln 1 (1 1 − 1 x).

From the rule ln 1 x

n

1 = 1 n 1 ln 1 x, it follows that ln 1 (1 1 + 1 x)

− 1

1 = 1 −ln 1 (1 1 + 1 x). Then

ln ln ln( ) ln ln+= ×= =l 24 24 8nn( ) ln ln ln 30

ln ln ln ln

235 2 3 5

63

6

3

2

×× = + +

−= = ln ln ln ln ln

l

232222

3

==++

nn ln ln

1

2

22

1

==−

−

lln ln ln ln ln

ln ln( ) ln

1

2

1202 2

1

3

27 27 3

13

=− =− =−

== ln ln−= =ln

−

25 5

1

25

2

ln ln lnxy

x

y

−=