Cambridge Additional Mathematics

Logarithms (Chapter 5) 135

Discussion

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We have seen that

p

2 cannot be written in the form

p

q

where p,q 2 Z, q 6 =0. We therefore say

p

2 is irrational.

More generally,

p

ais only rational ifais a perfect square.

What about logarithms? The following is a proof that log 23 is irrational.

Proof: If log 23 is rational, then log 2 3=

p

q

) 3=2

p

q

) 3 q=2p

The left hand side is always odd, and the right hand side is always even, so

the statement is impossible.

Hence log 23 must be irrational.

Under what circumstances will logab be rational?

Discovery The laws of logarithms

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What to do:

1 Use your calculator to find:

a lg 2 + lg 3 b lg 3 + lg 7 c lg 4 + lg 20

d lg 6 e lg 21 f lg 80

From your answers, suggest a possible simplification for lga+lgb.

2 Use your calculator to find:

a lg 6¡lg 2 b lg 12¡lg 3 c lg 3¡lg 5

d lg 3 e lg 4 f lg(0:6)

From your answers, suggest a possible simplification for lga¡lgb.

3 Use your calculator to find:

a 3lg2 b 2lg5 c ¡4lg3

d lg(2^3 ) e lg(5^2 ) f lg(3¡^4 )

From your answers, suggest a possible simplification for nlga.

From theDiscovery, you should have found the three importantlaws of logarithms:

IfAandBare both

positive then:

² lgA+lgB=lg(AB)

² lgA¡lgB=lg

μ

A

B

¶

² nlgA=lg(An)

C Laws of logarithms

where p,q 2 Z, q 6 =0

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Y:\HAESE\CAM4037\CamAdd_05\135CamAdd_05.cdr Friday, 20 December 2013 11:58:26 AM BRIAN