Cambridge International Mathematics

Challenge

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1 Where is the error in the following argument?

1

2 >

1

4

) log(^12 )>log(^14 )

) log(^12 )>log(^12 )^2

) log(^12 )>2 log(^12 )

) 1 > 2 fdividing both sides bylog(^12 )g

2 Solve forx:

a 4 x¡ 2 x+3+15=0 Hint: Let 2 x=m, say.

b logx= 5 log 2¡log(x+4).

3aFind the solution of 2 x=3 to the full extent of your calculator’s display.

b The solution of this equation is not a rational number, so it is irrational.

Consequently its decimal expansion is infinitely long and neither terminates nor recurs. Copy

and complete the following argument which proves that the solution of 2 x=3 is irrational

without looking at the decimal expansion.

Proof:

Assume that the solution of 2 x=3is rational.

(The opposite of what we are trying to prove.)

) there exist positive integerspandqsuch that x=

p

q

, q 6 =0

Thus 2

pq

=3

) 2 p=::::::

and this is impossible as the LHS is ...... and the RHS is ...... no matter what valuespandq

may take.

Clearly, we have a contradiction and so the original assumption is incorrect.

Consequently, the solution of 2 x=3 is ......

4 Prove that:

a the solution of 3 x=4 is irrational

b the exact value of log 25 is irrational.

638 Logarithms (Chapter 31)

IGCSE01

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Y:\HAESE\IGCSE01\IG01_31\638IGCSE01_31.cdr Tuesday, 4 November 2008 12:05:20 PM PETER