Logarithms (Chapter 5) 133
8 Findxif:
a lgx=2 b lgx=1 c lgx=0
d lgx=¡ 1 e lgx=^12 f lgx=¡^12
g lgx=4 h lgx=¡ 5 i lgx¼ 0 : 8351
j lgx¼ 2 : 1457 k lgx¼¡ 1 : 378 l lgx¼¡ 3 : 1997
In the previous section we defined logarithms in base 10 as the
inverse of the exponential function f(x)=10x.
If f(x)=10x then f¡^1 (x) = log 10 x.
We can use the same principle to define logarithms in other
bases:
If f(x)=ax then f¡^1 (x) = logax.
If b=ax, a 6 =1, a> 0 , we say thatxis thelogarithm in basea,ofb,
and that b=ax , x= logab, b> 0.
b=ax , x= logab is read as “b=ax if and only if x= logab”.
It is a short way of writing:
“if b=ax then x= logab, and if x= logab then b=ax”.
b=ax and x= logab areequivalentorinterchangeablestatements.
For example:
² 8=2^3 means that 3 = log 28 and vice versa.
² log 5 25 = 2 means that 25 = 5^2 and vice versa.
If y=ax then x= logay, and so
If x=ay then y= logax, and so
x= logaax.
x=alogax provided x> 0.
Example 5 Self Tutor
a Write an equivalent exponential equation for log 10 1000 = 3.
b Write an equivalent logarithmic equation for 34 =81.
a From log 10 1000 = 3 we deduce that 103 = 1000.
b From 34 =81we deduce that log 3 81 = 4.
B LOGARITHMS IN BASE a
logz is the power
that must be raised
to in order to get.
b
a
b
y=x
O
y
x
1
1
(1 a),
(a 1),
f(x) = ax
f-1(x) =logzx
4037 Cambridge
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