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Proof : We know, logaM logbMulogab [formula 5]

Putting M a we get,
logaa logbaulogab

or 1 logbaulogbb;

b

a

a

b log

1
?log

or
a

b

b

a log

1

log (proved).

Example 7. Find the value : (a) log 10100 (b) ̧
¹

·

̈

©

§

9

1

log 3 (c) log 381

Solution:
(a) log 10100 log 10102 2 log 1010 [log 10 Mr rlog 10 M]
2 u 1 [logaa 1 ] = 2

(b) log 3 2 log 3 [ log log ]
3

1

log

9

1

log 3 3 2 ̧ 3 2  3 aMr r aM

¹

·

̈

©

§

̧

¹

·

̈

©

§  

 2 u 1 [logaa 1 ] = 2

(c)

8

3

24

3

4

log 381 log 33 log {( 3 )} log 3

8

8 1 ,[ log 1 ]

8 log 3 3 [ log log ]

u

a

M r M

a

a

r

a





Example 8. (a) What is the log of 5 5 to the base 5?
(b) log 400 4 ; what is the base?
Solution : (a) 5 5 to the base 5

2

3

1 ,[ log 1 ]

2

3

log 5 ,[ log log ]

2

3

log 5 5 log( 5 5 ) log 5

5

2

3

5

2

1

5 5

u

u

a

M r M

a

a

r

a





(b) Let the base be a.
? by the question, loga 400 4
?a^4 400