ページ1:

x = x ax = b <=> x = log₁ b
log 10 = 1 log 1=0 log10x = x 10/09 -
a^=6<=>x=10gab
1. (1) 2² = 8 => 109 28 = 3
第2章 指數與對數函數
R) 5² = 125 => /og = 125=3
131) 1092 + = -3=> 23+
(4)log 0.001=-3(=> 10-3 = 0-001
对权有意義 ①真b20 ②底权a>0 ③底权ax1
2.若对蚁10gam1 (5-2X)有意義,則x的範圍?
logx-1
①真奴5-2x20
②底权[X-120
4x-1=1
<=
①② <x<x≠2
K<X
logal = loga ax = x alogax
loga G=1
3.11) 10g 10000-4
log 0.0001=-4
12) 3/093√2 = √2
loya X
xx loga a=X
=
laga a
= x
=X
= x
logars = logar +logas logas = logar-logas logar" => "logar
(1) log 354 +10936-21932 (2) lag 63-2logb+5log 2 - log 1443
4.
= log 3 (54x6) - 10932²
27
= 1093 154 x83)
= 109381
= log 334
= 4 x 10933
= log(62×32425.
36
=log/00
=2#
+4
17

ページ2:

Togas
C7O
C#1
Toge
log 2 = 0.3010
範例
『前國際例
5.10923 =?
log₂3 = log103
logo 2
log 3 = 0.4771
0.4771
=1.5850 log=0.8451
0.3010
15=0-6990
n# 0
6. lega blogab minER
=
(1)/09324 = 109332² = = 1992 2 = 1/1/11
(2) log | 2 = logo-32 = = = x/0922 = = 1/12/1
logab log₁ α = 1 => logab = 10
logia
7. (log = 3 + log49) (log 3 4 + logą 2)
3 #
2')
= (log = 3 + log = 3)|log 3 2² + log 3?
= (log2} + \ / 10923 ) ( 2log = 2 + ½ log; 2)
Er) = (2x1023)( {x10932)
某次地
試問芮]
=
2 x x log₂ 3 x 10g 32
倍?(整
代入
41
=
1
= 1× × 1 = 5
logablogi C x loged = logad
= 7. 10923 logs 49 x 109325 x logn4
Ecg
= logz 3 x logs 7² x log 35 x logn 22
=log23
= log 23 x 2log 57 x 210g35 x 2logn 2
=
2x2x2 x 10923 x 10g 35 x logs 7 x logn²
log
= 8 x log₂ 2
=8

ページ3:

,
log f(x) = logg (x) => (fax) > 0
{g(x)>0
(fou = g(x)
9. Blago (x-2)+1096 (x-3)= 1
真双70
(1)真5x-220
底权>0
权x-3>0
1x23
底权$1
(2)对权相加二真奴相乘
1096 [(x-2) (x-3)] = 1 =10966
= (x-2)(x-3) =x5x76-b 725 XE
x(x-5)=0
RE
X=0/50
#
1. ^^ = log = N= b = log = 3 #81ª + 26 = ?
3² = √5
b
√√√² = 3
81+26
2
=(√√5)+(3)²
=25+9=34#
2. alog 520 2+ blog 520 5+ Clog52013=3 $0₁+ b + C = ?
Blog sao 20 logo
sloga
=
a
x1+x*+ = 109520 (2ª × 5 × 13°)=3 logars=logar + logas
真相x
a
5203 = 2ª × 5 × 13°
51520
21104
2152
2126
73
2³×5×13
=12³×5×1
(2³× 5 × 13)³
= 2² × 53 × 13³
a+b+c
=9+3+3
= 15
#

ページ4:

log 108 = log (2² x 3")
3. log 108 = a log 24 = b xa-blog 22 #log 2 = ?
a=2log 2+3log 3-0
= 2log 2+3 log 3
log 24 = log (2³ x 3)
-6=3log 2+ log3 - 0
②x3-①消掉1093
2
=
·Blog 2 + log3
3b-a=9log 2+ 3log3-2log 2-3tag 3
=7log2 log2=3b-a
4. log₂ 3 = a log37=b 1xa.blog4256 = ?
log4256
=
109256
109242
=
=
310g 22 +ab
1+a+ab
3+ab
Tatab#
#
1092(2³×7)
Toga (2x3x) = logz 3x log = 7
= log₂²² + Tog. 7)
=
saxb
Tog 2 + logz 3 + logz
↓
↓
a
5. 解方程式 9x+2x3x+1-16=0
目標:一元二次方程式
13²)x + 2x 3 x 3'-16=0
(3x)² + 6x3x-16=0
(3-2)(3x+8)=0
3x=2 => x = log32
- #
3x=-8 => X 115 BX>0)
6. AF = log x 100 - log x + 1=0
logx 100 = logx 10² = 2109x10
log x 10 A log, x ₤1 BU EX
logab x log, a = 1
3 log10 x = + logx 10 = =
1 = log x 100-logx+1
=2/0g x 10-logx+1
=2x+-++1
t
#
= -++1 X = 10 1100
5×+ => 2-+²++= 0 ↑
(t+1) (t-2)=0 += -1 = log₁ox

ページ5:

stag}"
對數函數及其圖形
y=f(x)=10gax稱為以D、為底的对权的权0、200x1
1. f(x)=10g5x y=f(x)=logax
(1)f(25)=5
(2)f(1) = 0
(3)f(六)=-2
(1)
x
(2)
11,0)
*為正實权
a>1 嚴格遞增
OKOKI 嚴格遞減
2.求作y=f(x)=10g2x的图形
2
x
12
4
2
+
>x
2
y-2-10
2
1
4
2.
a71 凹口向下
0<a<1凹口向上
*圖形恒在由右方→X值為正
*y軸為漸近線
3.11€ = log X
1
y-2-1
x
9
37
3
2
0
2
39
y=logzx
y=log3x
→
112345678910
*底权越大
y=logax 與loga(-x)的圖形对稱於軸
y=logax與-10gax=10gX的圖形对稱於X軸
4. 1Ey=log₂ X y = log(x) -y=logaα D
y
y=logz1-x)
y=log=x
y=log2X x|1
2
4
yo
2
→x
越右越低
-y=log2X
x111214
y=10g2(-x))→真权-x>0
=y=log₂X x|-11-21-4
81-11-2-4
X40

ページ6:

y= logoXh1 => y = logo (x-h)
y = loga X ±4k 1 = y = k + loga X
1111 y=log = (x-1) y=log = x+2y=log₂ X by
→右移
>上移
y= log₂ x
2/6/7/2/2
(上)
(右)
y = log XY
0
12) y= log₂ x 548 31
=> Y=log₂ (x-3)
再下移5單位
=>4=1092(x-3)-5
f(x)=g(x)的實权解個权⇒y=f(x)與y=g(x)兩图形交点的個數
6. log X = X- | 10?txh} ?
Y
21
y=1092x
x1112141
y=x-1
X12
Yo
271
loga X>0
演練
序
a>
10 <x<\ => loga X<0
7.哪些对权值>02
|0<a<\/ x> => loga x<0
0<x<1 => 109 x>0
(A)log;} {Bilog } } (Cology! (Dilog1} {{) lot'}}
a> |
E
贈
X31 Y>00
X21720
13 (E) 13
LAE)
02021
X<14>0
X>Y<0

ページ7:

對數函數及其圖形
qxxl f(x)=logax為嚴格遞增函权⇒X值愈大,权值就愈大
0<9<l
減
=>
8. A = log + 5 b = log + 2 C=log₁ = d = log₁ = **N'?
小
1.判斷底权
2.x:5>2>3>
3> 遞減
ya<b<c<d
指权时权與对权函权互為反出权 f(x)是f(x)的反函数
9.f(x)=ax+b通过(17) f-'(x)通过(4,0) 权对(a,b)=?
f(1)=a+b=7
> 反函數:xy互換
f(x)通过(0,4)
f(0) = 0+6=4
V
a+b=7 α=4
=> (a,b) = 14, 3) #
1+6=4→ b =3
对权不等式
10. 7 logz (4-x) = 1 + logo (x-1)
真权>0
step1. [4-x20→x24
>>1444
底权>0
1< x < 0< 1-X
底≠ 1
step 2. log₂ (4-x) = log 4 4 + log4 (X-1)
→ log₁₂ (4-x)² = log₁ (4 × (x-1)]
(換底公式)
(logars = logar + loga>)
step3. ∵遞增
(4-x)² = 4 (x-1)
16-894x224X-4
x²-12x+2020
(X-10)(x-2)20
大於在兩辺 小於在中間
7210
X≤2 x 1X24
找交集
2017.
由
FKX52
2
4
10

ページ8:

对对权函數轉換成多項式的权,求Max/min
11.xz / B fix ) = log (x 1)-3 logxx = ? f(x) Famin =?
fix)= log (xlogx) - 3 log x²
= logxx logx - blogx
= log X)² - blog x
=
+2-bt
=(4-3)2-9
t = 3 = log x => x = 10² = 1000# min=-9#
ab.c.dŔ0.1 *N
1. 判斷a.
Y
y=loga X
-y=leg₁₂x
>x
=y=logcx
a.b>1
a<b
b>a>lc>d>0#
-Y=logy x occidet d<c
2.下圖為y=a+10gbxa、b均為常权何者為真?
(A) D.<o b>1 (B) a>o, b>1 (c) α=o, b>1 (DI > 0<b<l
T
投
19
(E) <0.0<b<l
逐
『贈
E
Y
以下移:9<0
1234
>x
2.0<b<1
IE?#
3.求下列方程式的實权解個权
111x-1=10921×1
314
(2) X-1=1logx|
2個#
91,0)
Y = X-1 X10
7/12/0
y=log=1x1
y=x-1(同左)
870
y=1100=xX1>0

ページ9:

4. ½ log = (log = (x) >I
1. 真奴x>0
2
[
X的範圍為?
log, X>0x1
· log, lloga XXS log₁ =
=> 271
注意大於小於
log, x © +
→ log X © log¸ z² => X © 2³
J. K<x<₂ #
2-4 指权與对权函权的應用
科學記號 a=bx107 n是整权
146410
1. 11) 539000 = 5.39 × 105
(2) 0.000539 = 5.39 × 10-4
"Exα= bx 10" ntx 1≤ b < 10 01/09α = n + logb
n是整权
n
Allogα
(1) loga的首权: (2)10ga的尾數:10g0
2. 已知10g3.561=0.5516則10g0.0003561的首权?尾权?
log 10.00035611
= log (3.561 × 10-4)
= log 3.561 + log 10-4
=0.5516-4
2 = -4
尾数=0.5516
→小权点後第4位開始不為0
loga的首权為n<=>a的整权部分為n+1位权(當a>1)
3.(1)已知10ga=3.5520G的整數部分是?位权
log4=3.5520
=3+0.5520
3+1=4位权#
(2)已知10g20-3010 226是?位权
226 => /og 226
=26xlog 2
=26x0-3010
=7.826
7+1=8位权#
"

ページ10:

自小双点後來首位開始出识弟
(30<^<1)
4.1110ga=-5.43210.在小权点後第2位開始不為。
log α = -5.4321
=-6+0.5679
=
第6位不為01
→屋权寫正
5. 將2-18化成小权,小权点後有連續
(2)將(3) 20化成小权,其小权点後第?位開始
1-4130 - 3-30
=
10g3-30 = -30x10g3
=-30x0.4771
不為口
14
n個0,第一個不為口的數字為
log 2-18=-18 x log2
a
=-18×0-3010
=-5.418
=-6+0.582
第6位不為1022
= -14.313
=-15+0.687
第15位不為0~
→尾权寫正
本金P每期利率r 期权n
→尾权寫正 「單利: n期後本利和Sn=P(1+nr)
0.4771<0.582 -0.60201 複利:n期後本利和Sn=P(1+r)^
↓
log 3
log 3. ~ log4
:(na)=15,3)
7. 冰原覆蓋率(A%)
開始融冰後經過時間(七年)
A = 100 × 12/11 -k+
1000前~now(80%)
80=100x()-kx1000
⇒(音)-1000k=80=生
now ~ 1000% (?%)
100
A=100x()-kx20
⇒(1)-1000k)2
5
⇒100×1号产=100×8=64%
6.年利率12.5% 複利至少超过2年
本利和才会超过本金的2倍
Y=12.5% 設本金x
7年
x(1+r)">24
(1+12.5%)722
100
+100)=112.5
-12">2
100=4-5
==> log (1) " > log 2
⇒n(10g9-10g8)>10g2
↓
↓
23
log 3² log 2³
=2log 3 = 3log 2
N
=
4
9.
8
⇒n(2x0.4771-3×0.3010)>0.3010
⇒n10.9542-0.9030)>0.3010
⇒n(0.0512)>0.3010
n=0-3010
0.0512
= = 5.879 (nπ186)