For example, since 8=2^3 we can write 3 = log 28. The two statements ‘ 2 to the power 3 equals 8 ’
and ‘the logarithm of 8 in base 2 equals 3 ’ areequivalent, and we write:23 =8,log 2 8=3
Further examples are: 103 = 1000,log 10 1000 = 3
32 =9,log 3 9=2
4(^12)
=2,log 4 2=^12
The symbol,
“is equivalent to”
In general, y=ax and x= logay areequivalentstatements
and we write y=ax,x= logay.
Example 1 Self Tutor
Write an equivalent:
a logarithmic statement for 25 =32 b exponential statement for log 4 64 = 3:
a 25 =32is equivalent to log 2 32 = 5.
So, 25 =32,log 2 32 = 5.
b log 4 64 = 3 is equivalent to 43 =64.
So, log 4 64 = 3, 43 =64.
Example 2 Self Tutor
Find the value of log 381 :
) 3 x=81
) 3 x=3^4
) x=4
) log 3 81 = 4
EXERCISE 31A
1 Write an equivalent logarithmic statement for:
a 22 =4 b 42 =16 c 32 =9 d 53 = 125
e 104 = 10 000 f 7 ¡^1 =^17 g 3 ¡^3 = 271 h 27
1
(^3) =3
i 5 ¡^2 = 251 j 2 ¡
(^12)
=p^12 k 4
p
2=2^2 :^5 l 0 :001 = 10¡^3
2 Write an equivalent exponential statement for:
a log 2 8=3 b log 2 1=0 c log 2
¡ 1
2
¢
=¡ 1 d log 2
p
2=^12
e log 2
³
p^1
2
́
=¡^12 f logp 2 2=2 g logp 3 9=4 h log 9 3=^12
3 Without using a calculator, find the value of:
a log 10100 b log 28 c log 33 d log 41
e log 5125 f log 5 (0:2) g log 100 : 001 h log 2128
i log 2
¡ 1
2
¢
j log 3
¡ 1
9
¢
k log 2 (
p
2) l log 2
¡p
8
¢
The logarithm of in base
is the exponent or power
of which gives.
81
3
381
Let log 3 81 =x
626 Logarithms (Chapter 31)
IGCSE01
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Y:\HAESE\IGCSE01\IG01_31\626IGCSE01_31.CDR Monday, 27 October 2008 3:01:48 PM PETER