五期
(2) A surd of any order may be transfumed inta
a sund of a
PENETAKKY
WAH KIU YAT PO
# B
1768
#
英文
rey
数單科
(109)
你您絡交
different order
Surds of different anders may be transformant
into surd's of the same avdler.
} (4) (ab)* *a*. 6x
i's
A surd which is expressed so that the infegei under the root sign is as small as possible said to be in its simplest fo
form:/
MATHEMATICS (14)
LESSON 14 INDICES AND LOGARITHM
§A, Indices and Surds:
¿B, LOGARIZHM:
We are familiar with the following laws i
<i>
a" ÷ a" =
am-n
(2")"= Q.
Ya* = (Ya} .fro
aq o
can be expressed in the form A“, s.4. AT=N;
If number N then, X is called the logarithm of the no. N, to base 0. NATATION :
X une
Logan
The following are the main properties of logaritheo . If a is a non-negative number such that ask 1, then
at is always positive, for all x ( x may be -'me, o or 've). Hence, it is meaningless to say log. of a
-'re number
log. of zero.
</a>
af
a
= Va
770
J
2
EXAMPLE 1 Simplify and express with positive indices:
(a) 524 (517-2
3,
loga! Log, a=!
= 0
*
Log (MN)=
・log. M + log!
3
(0) 1.4 20 125 ÷ 2.5
SOLUTION A
<a>
Exp. = 5
*
5
<b>
Exp.
= 5 #5454
-3
5*f
பதி
20
for any non - zero
a.
5, loga () = logam ~logan.
*; Log. (N*=* Log.M; for any/h (+'ve, - 've, a, faction)
Log A/N == log. N
er integer
NOTE: <> Log, MI log. Not tog (mzm)
<2>
ley Log M
-alog.n
<3> loom + logo M-
Log
das from property 5, when Mul, Log = log. Na cologn
<5> Logarithm to base to is called commen logarithm. NOTATION: Log, Nal
more general loz N=x
which is most used if practical calculation,
was introduced by English
and
ematician
*RARE!
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今秋開始實施
**
器
能加
備有
Ext
0. if xo.
3
-(#)×(÷) + (2±)
= 2.71240
Natural
retical mathewmedier.
Mahleaticas.
log49)
Wher
is called
ogarithm
() () (룿)
x
- () () ()
-
Example &· if x = a
+
(~1)
prove that *40 (44* - 3)
and cut Napit
"EXAMPLE 2: Simplif
SOLUTION (*** Exp.
Prest
22000 A+
I" (a - J.
3
HONG
Example 21 SOLUTION
2
— za3 + 2(32)√ā**7)*
2 Q3 + 6 a { a2-1)
=20(40*-34
Solve √+6 +/+ = √2+7
{√x+6 +/+) =
62 + 7
6 x + 7 4X
←
(x+2) ( x − ) = 0
Checking, (a, when
X - 3
x
1258-118-1
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[雙百元,三批五十
-
<b>
HINTS & AM. To E11
az (62 - 43*
«*» (a*~¿” + abi a*- 4* — ab)
«e» (X+1) 1293) EX ̃– $2+ Q)
<f> 20°($Q+ 4) { 30−2} {#Q*++).
«J» (x=7) (2x+550x-2)
Noth Apply Remainder Themem do 1974
10 - 21x-10= (52+2}{42 −5)
het F(x)=100x^+ 10x+p, tika F°° 3)=8&j= c
by Exp.
3(Log2)
Satlogs)
Log +21 (og 5)
($) + log # ̈‚„X”
(.log
(Long
(Loy
BRARY
10
* Log 10°
following relation.
gitar) + log, 4.
J
KOK
4. S. =/X+6 +/7+T
#√3+6 + £7
A 3 + 2 = 5
R.S. NG*3+7= 5
(b) when * --
~ 4. S. ==
-√5{ + SE1
-+-
ALLE, MA. S
SOLUTION.
-4+
R.S=√6(-4)+7 =B ---- LSAR.S A is NOT a solution of the equation © This shows the importance, in checking)
EXAMPLE *: Rationalise -
Saamzima Exp. =7
ANS
log (27+1) - loyal - Loga
σ
loyal-224) = log. 1=loy a
(
ay
A
uy arty
_{X*+1) ("*+a)= daly
EXAMPLE 7 : if a'r b=7a6, then log at b = 1⁄2 (loga + log bì
PRODE: 書
a2 + 5+ cab rab+zab
(a+b)= gab
#
kog (a+b)" = Lig(4ab)
Log (a+b) =
log at log b + log 4
or log(a+b)= ±( log at Log b) + 1 6og 9 Log (a + b) -
log 5=
3 = = { lag a + Lag b} log Att ! ( log a + . (og b)
J** *** **
< Ахау уакустика,
་
4-xy ther
(1) (y-1) = 0
But
2-1
*+ y +2 = x+xy = 1
Zie
4 = 1.
EXERCISA 14
Find the meaning and value of a", where a mo .
2, Simplify and express with positive indices ›
225 x 72" x 1000
<A>
3, 14 x
+
find the wahre of
5
EXAMPLE &
Solve
= 21.
nth
}
- √3 + E + !
2-274
4 - (242**
2.5+ 2/T + 4/+6
=-{176 + 5 + 25 +
perfect Pack (0) If A is a rational number and is not a
then I is called a sued of the power, nth order. ce.g. fe is a quadrats sund) Hence, surds are inexpressible wither as integen or as fraction, but the value of a surd can be obtaineït to any degree of accuracy.tn.g, mn festational number )
alstien: Take logarithms for both sides ( to base 10):
log (6*** 3** - *}m log 21
Log 21
(3x+1) Log 6+ (15x-2) log 3 =
log 21-logh + 2 logs ( 3 Log 6 + 15 Log 3) x =.
Log 21 - log b + 2 logs
3 ing to + 15 Log "
X =
43222-0:2782 +0.9542
3 x 2.77 +15 40.4477
LELEL
{ car to 2 Mac, ɔ
4 Given
m = 3 +2/,
7=8-25.
find
(A) MA
and (c) m'ent
5, Solve the following equations
"
- x
2
cb √42^+172 + 15
−√2x^ + 52- 3 = · 2 x *— 12x = 18; Given log 2 =0.30103, log 3 = 0.47712, obtain approximate values of the logarithms to base 10, of (as Log 2.5. (b) log 15 and (e) log 0.12
7. Solve
(log(x − y) + Log (7x - 8 y) = 2
Log (23+y+) = log(x^~29 + y2)m } .
$, Solve 0.93o m 1.832