26 Higher Engineering Mathematics
y
0.5
1.0
0
1
2
3
20.5
21.0
x
x
3
0.48
2
0.30
1
0
0.5
2 0.30
0.2
2 0.70
0.1
2 1.0
y 5 log 10 x
Figure 3.1
(ii) log a a = 1
Let log a a = x then a
x
= a from the definition of
a logarithm.
If a x = a then x = 1.
Hence log a a = 1. (Check with a calculator that
log 10 10 = 1 and log e e = 1)
y
2
1
0
1
2
3
4
5
6
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 20.69 21.61 22.30
21
22
y 5 log e x
Figure 3.2
(iii) log a 0 → −∞
Let log a 0 = x then a x = 0 from the definition of
a logarithm.
If a x = 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 log a 0 → −∞