Higher Engineering Mathematics, Sixth Edition

26 Higher Engineering Mathematics

y

0.5

1.0

0 123

2 0.5

2 1.0

x

x 3

0.48

2

0.30

1

0

0.5

2 0.30

0.2

2 0.70

0.1

y 5 log 10 x 2 1.0

Figure 3.1

(ii) logaa= 1

Let logaa=xthenax=afrom the definition of

a logarithm.

Ifax=athenx=1.

Hence logaa=1. (Check with a calculator that

log 1010 =1andlogee=1)

y

2

1

0 123456 x

x 6 5 4 3 2 1 0.5 0.2 0.1

1.79 1.61 1.39 1.10 0.69 0 2 0.69 2 1.61 2 2.30

21

22

y 5 logex

Figure 3.2

(iii) loga 0 →−∞

Let loga 0 =xthenax=0 from the definition of

a logarithm.

Ifax=0, and a is a positive real number,

then x must approach minus infinity. (For

example, check with a calculator, 2−^2 = 0 .25,

2 −^20 = 9. 54 × 10 −^7 ,2−^200 = 6. 22 × 10 −^61 ,and

so on)

Hence loga 0 →−∞