Cambridge Additional Mathematics

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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Y:\HAESE\CAM4037\CamAdd_05\133CamAdd_05.cdr Friday, 20 December 2013 1:04:39 PM BRIAN