1969-02-04 — Page 23

REFERENCE NIDHAA

★教堂 莫三第張六第日八十月二十年申戊暦夏 WAH KIU

YAT PO

CITY HALL

二期星 日四月二年九六九一公年八十五國民中

1969

試題預習

(3) In solving irrational equations,every"TOOS.

must be checked.by.substitution('in the origina lal equation).

"SOLIITION.

Tog

-108x-108

1-logaly + )+log y

數學科 (十四) 歐陽鋊文

loga (7)(2)] = log(+12); I

= logs th

EXAMPLE 4*

Express in the simplest

MATHEMATICS (14)1-

SON 14: INDICES AND LOGARITHU

A) INDICES AND SURDS:

We are familiar with the following laws:

ato

at = Ya3 = (1a) : 8+0

SOLUTION

肉味

EXAMPLE 1 Simplary and express

ve

EXAMPLE 5 Rationalise

EXAMPL

PROOE

1) a4axg

then le

$(log atloguj

a+b2ab7ab2ab

(a+b)

9ab

log(9ab)

loglatb

2.log(a+b) Inga+logb+log 9.

or log(a+b)=(log. a+log+log 9. 108(a+b)- 1083=+ (log a+ log b )

log

== (loga + log b

(a) (b_-)* (;;) =

6X3

*7

SOLUTION

La) Ext

3(3

SOLUTION:

5√-3

(5E) - (3/x)

Lo a tájaz 25:a-9x

Note: (1) If & is a rational number and is not a per-

fect no power, then Wk is called a surdl of the nth order. (e 62 is a quadratic sured) Hence, surda are inexpressible either as integers or as fraction, but the value c

of a surd can be obtained to any degree. of accuracy. (e.g. an irrational number).

A surd of any order may be transformed into

a surd of a different orderin

(3) Surda of different orders may ce transformed

into surds of the same order.

•4) (ab)^ — aħ bħ (5) A surd which

expressed so that the inte-i ger under the root sign is as small as poss- ible, is said to be in its simplest form.

(B) LOGARITHIS:

If number N can be expressed in the form a; then, x is called the logarithm of the N, to base at NOTATION

X1OF N

The following are the main properties of logrit

1) If a is a non-negative number such that a 1. ther A is always positive, for all x x may be ve orve). Hence, it is meaningless to say loz. of ave number or log. of zero. logal 0, for any non-zero log a

loga (MN) == log. M + log N

loge (~) = lag M-z logon

EXAMPLE 21 18 x

prove that 2a(4a* −3)

6, log(N")=k

for any

fraction og á ger]);

Log. BN + Loga N

EXAMPLE 9 Solve

SOLUTION: Take logarithms for both sides (to base 10):

108 (632H 315^- ̄) — 108 21

(3x+1)log 64(15x-2)10g - = log 211

(3 log 6+15. log3)x = log 21-10g642g3

log-21 log 6 + 2 log

سعار

3 log 6 + 15 log 3

1.3222 - 0.7782 + 0.9542

3x0:7782 + 15 x 0.497

14982

9.4911

240.16 (corr. to

HINTS & "ANS.

(1) F(-3)=55; F(k-1)=

F(2t-3)=(6t−5)(4t-18)

(2) G(2,-2)=16} {(x+1, 9+1}={(x+1)=(y+1)] = (x− y)2

G(-2g, -38)=(-26 +38)*=

4) Let F(x) 3x+x ax®+5x+b

Then RF(1)9÷atb

RF

5)(a) 2xy(xy-3}{xy+1) ̧

(b) 3x(5x-4)*

(c) (x2-7° #L: +m" } {x′′-y^-

(d) (a-b2 tab)(à" -b-ab) (e) (x+1)(x+3)(x2 -3x+a)

(1) 2a (3a+2)(3a-2)(4+1))

(8) (3x-7)(2x+5) (x-2)"

(h) (x2+2x+3)(x2+2x+3),

(1) No shorter form

(1) (hax-b)(bx-la)...?"

Note: Apply Remainder Theorem to (e),(f) and (g);

(2) (4-0)® -(cta)k, since

d) even

-(cd) kodd

And

EXAMPLE

SOLUTION:

2a+2(38)(451)

(2)

tat

2a

(x+6)4(x+1)+26(x+6)(x+1)

x2+ 7x+6)

Checking a when x=3,

1.5

(b) whe

2a4a-3

6x+? 6x+7

5--16 +/- T

=155 15

Note:7(1)]log, Mlog&N log(M‡ N)

(2) 10g. # # 1000M

loga #*

LOERE loga M loga N

(4)

*5**when' 1

·logat=

fog. Na co log N (5) Logarithm to base 10.is.called common

logarithmes

NOTATION: 10g NX or more general, log N=X

which is most used in practical calculation and was introduced by English Mathematician Briggs:( 1556 1631)

(6) Logarithm

Where-e=1

base.

is called Natural logarithm:

(2.71828 18234-

Natural log.

is usually used in theoretical mathematics, and was intrduced 4 by English Mathematician

Napie ( 1550

– 1617)

EXAMPLE 61 Simplify (a): 2 log-logz +2 10g 3+10849 (b) (log_2}+(log_5)+(log5 )(1088)

SOLUTION: ( & )EXP --- log(§)-log — +log 3′′ +log.

-

tog ( = = 2 × 7 × 9 × 7J 108-100

2

-21x-10=(5x+2)(2x-5),

{Let{F(x)=100x +10x+p [then F(-\ )=N{ =)=0

[?) [~xw+2=★+++ = = 1}

[Wyz4zx+xy ➡ xyz.

ANS. 4 or 150

or z{x+y)+xy(1-2)= 0,′′ but*x+y=1-2

• {z+xy){1=2) — 0

zxy then

• (x-1){ \~1 ) == 0

ZAJAZ

EXERCISE 14

1) Find the meaning and value of a

where a 40)

Simplify and express with positive indices: (a) 225 x 72 x 1000-

(b) (x***+%

(3) If x2 + ×

*$3,ìnnaithe value of

==√6(~$)+] [=√3] JL.ST#R.S.

is not raysolution or the equation This shows, the importance, in checking)

x=31 Ans

Note: (1) In the Tabove example (or other examplesfof

the same kind), the position value of the square root is always taken. Also all the are supposed to be real.

(2) From the above example, we find that when we

square the equation, the resulting equation is not an equivalent one, which will contain all the original equation, if any but it may happen that some of the roote v theforiginal equation.

t satisfy

(b)EXP.

— (log2)+(1085) +(log5)(log2")

(10g2+log5) ((log2)-(log2)(log5)

+(log5)+3(log5)(Log2) [Log(2x5)] [(10g) -{log2)(1085) +(log5)] +3(log2)(log5) ➡ (10g2)+2(log2)(log5)+(log5)`

(log 2+ log 5 )* (log 10 )* -1

Note: (log 10) # 10g 10

where (1oz 10) — (1)2 = 1; 10g 10-2 log10-2

EXAMPLE 7: Find xy which satisfy the following re-)

lations

102. (x2 +1)-10g, x-log 4—1-log (y*+e*)+10%a7

47: Given me 3+2 42, n=3-2/2, find

(a) mn,

and

.5) Solve the following equations:

(a) } //* + 8 -'x=2

(8)§44x2 + 17x + 15}~4/2

√-2x2-12x-18

6)Given Log 2 = 0.30103; 1og 3=0.47712, obtain,

approimate values of the logarithms to base 10, for (a) log 2.5(b) log 15 and (c). log, 0.12

[ log(x+y) + log (7x=8y)=2

{ log(x-y3)i- log (x^-894y=) ==|z

(8)[Solve10.931.832

(7) Solvej