Page 123 - DMTH401_REAL ANALYSIS

Unit 9: Functions

Notes
Figure 9.10

Logarithmic function to the base a has the following properties:
(i) log, (x x ) = log, X + log, X
1 2 1 2
é x ù
(ii) log a ê 1  = log x – log x . 2
a
1
a
ë x 2 û
m
(iii) log, x = m log, x for every integer m.
(iv) log = I.
a
a
(v) log = 0
1
a
By the definition of log x, we see that log 1 = 0 and as x becomes larger and larger, the area
1
covered by the curve f(t) = , X-axis and the ordinates at 1 and x, becomes larger and larger. Its
t
graph is as shown in the Figure 9.11,
Figure 9.11

You already know what is meant, by inverse of s function. You had also seen in Unit 1 that if f is
1 – 1 and onto, then f is invertible. Let us apply that study to logarithmic function.

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