單元 24: 對數函數

單元 _24: 對數函數

( {… § 5.2)

q _{b > 0} / _{b 6= 1.} Nbj˙

b^y = x, x > 0

íj _y 4uU)b _b í冪Ÿ _(power) Ñ _x ví冪 Ÿ_, 1J

log_b x [ý_, ¹

y = log_b x J/ñJ _{x = b}^y_{, x > 0}

/˚TJ _b Ñí _x íúb (logarithm of x to the base b). Wà_, 根Wúbíì2_,

(a) log₁₀ 100 = 2 (ÄÑ _{100 = 10}²₎ (b) log₅ 125 = 3 (ÄÑ _{125 = 5}³₎ (c) log_{3 27}¹ = −3 (ÄÑ ¹

27 = 3⁻³) (d) log₂₀ 20 = 1 (ÄÑ _{20 = 20}¹₎ (e) log1

2

16 = −4 (ÄÑ _{16 =} ¹

2

−4)

(f ) log 1 10

0.01 = 2 (ÄÑ _{0.01 =} ^{ 1}

10

₂

)

註 _.

b^y = x > 0 恆Ñ£_, ]c5?£bíúb

log_b x, x > 0

W _1.

(a) log₃ x = 4

(b) log₁₆ 4 = x

(c) log_x8 = 3

(d) log₁

2

4 = x

<j_{> (a)} âúbíì2_,

x = 3⁴ = 81

(b) âì21“_,

4 = 16^x = (4²)^x = 4^2x ] _{x =} ¹

2.

(c) âì21“_,

x³ = 8 = 2³ ] _{x = 2.}

(d) 根Wì21“_,

1 2

_x

= 4 = 2² =

1 2

−2

] _{x = −2.}

ù_泛Uàíúb系$Ñ

1. J ₁₀ Ñbíúb_, ˚Tàúb _(common logarithm) 1pA _log, ¹

log x = log₁₀ x, x > 0

2. JÌÜb _e Ñbíúb_, ˚TAÍúb _(natural logarithm) 1pA _ln, ¹

ln x = log_e x, x > 0

Éúbíl_, 藉Œk

úb律

1. log_b mn = log_b m + log_b n

2. log_b m

n = log_b m − log_b n 3. log_b mⁿ = n log_b m

4. log_b 1 = 0

5. log_b b = 1

註 _1.

log_b(m + n) 6= log_b m · log_b n

C

log_b(m − n) 6= log_b m log_b n

ÇÛ·<íu_, úbí D}.k D}íú b_, ¹

log_b m · log_b n 6= log_b mn C

log_b m

log_b n 6= log_b m n

註 _2.

mⁿ = log_b 1 − log_b mⁿ

= 0 − n log_b m = −n log_b m Ĥ_, úL<õb _r,

log_b m^r = r log_b m

Wà_,

(a) log(2 · 3) = log 2 + log 3

(b) ln 5

3 = ln 5 − ln 3 (c) log₃ √

7 = 1

2 log₃ 7 (d) log₅ 1 = 0

(e) log₄₅ 45 = 1 (f ) log₇ 1

√5 = −1

2 log₇ 5

W _2.

log 2 ≈ 0.3010 D

log 3 ≈ 0.4771 J£

log 5 ≈ 0.6990 t°-®íM_.

(a) log(15)

(b) log 7.5

(c) log 81

(d) log 50

(e) log 1 300

<j_{> (a)} â _{15 = 3 · 5} £úb律¶_, ) log 15 = log 3 + log 5

≈ 0.4771 + 0.6990

= 1.1761 (b) ÄÑ

7.5 = 15

2 = 3 · 5 2 ]âúb律_,

log 7.5 = log 3 + log 5 − log 2

≈ 0.4771 + 0.6990 − 0.3010

= 0.8751

(c) ÄÑ _{81 = 3}⁴_, ]âúb律_, )

log 81 = 4 log 3 ≈ 4(0.4771) = 1.9084 (d) â _{50 = 5 · 10} Dúb律, )

log 50 = log 5 + log 10

≈ 0.6990 + 1 = 1.6990 (e) â _{300 = 3 · 10}² Dúb律_,

log 1

300 = − log 300 = −(log 3 + 2 log 10)

≈ −(0.4771 + 2) = −2.4771

W _3.

(a) log₃ x²y³

(b) log₂ x² + 1 2^x

(c) ln x²

q

x² − 1 e^x

(d) ln x²

√x(1 + x)²

(e) ln xe^−x2

<j_{> (a)} âúb律_, )

log₃ x²y³ = log₃ x² + log₃ y³

= 2 log₃ x + 3 log₃ y

(b) âúb律_, ) log₂ x² + 1

2^x = log₂(x² + 1) − log₂ 2^x

= log₂(x² + 1) − x

(c) âúb律_, ) ln x²

q

x² − 1

e^x = 2 ln x + 1

2 ln(x² − 1) − x (d) 根Wúb律_, )

ln x²

√x(1 + x)² = 2 ln x − 1

2 ln x − 2 ln(1 + x)

(e) 根Wúb律_,

ln xe^−x2 = ln x + (−x²) ln e = ln x − x² 根Wúbíì2_, )úbƒbí

ì2 _.

f (x) ^def= log_b x, x > 0

˚TJ _b Ñbíúbƒb (logarithmic function with base b).

âúbíì2_,

log_b u = v J/ñJ _b^v = u, u > 0

¹_, õ _{(u, v)} Êúbƒb

y = log_b x

íÇ$,J/ñJõ _{(v, u)} ÊNbƒb y = b^x

íÇ$,_. ¢õ _{(u, v)} Dõ _{(v, u)} Jò( _{y = x} Ñ 鏡射_, 6ÿuz_, úbƒb

y = log_b x

íÇ$DNbƒb

y = b^x íÇ$Jò( _{y = x} Ñ鏡射_.

Ĥ_, Î7描õ¶Õ_, ª根W鏡射¶_, øNbƒbíÇ$ú ò( _{y = x} 鏡射_, )úbƒbíÇ$_, àÇý_.

根WÇý_, )

úbƒbí4” _.

y = log_b x, (b > 0, b 6= 1), x > 0

à-í4”_.

1. ì2域Ñ _{(0, ∞).}

2. M域Ñ _{(−∞, ∞).}

3. ¬õ _{(1, 0),} ¹ _{logb 1 = 0.}

4. Ê _{(0, ∞)} ,©/_, ¹恆©/_.

5. Jb _{b > 1,} Ê _{(0, ∞)} ,遞Ó_, ¹恆遞Ó_; J

b 0 < b < 1, Ê _{(0, ∞)} ,遞Á_, ¹恆遞Á_.

6. Jb _{b > 1,} † lim

x→0⁺

log_b x = −∞ / _lim

x→∞ log_b x = ∞ Jb 0 < b < 1, †

lim

x→0⁺

log_b = ∞ / _lim

x→∞ log_b x = −∞

âúbíì2_, )

Nú逆4 _.

1. ø般b _{b > 0} / _{b 6= 1,}

b^{logb x} = x, x > 0 /

log_b b^x = x, −∞ < x < ∞

2. AÍNbƒbDAÍúbƒb_,

e^{ln x} = x, x > 0

/

ln e^x = x, −∞ < x < ∞

註 _.

f (x) = b^x, −∞ < x < ∞ /

g(x) = log_b x, x > 0

†根W¯Aƒbí«d†DNú逆4_,

(f ◦ g)(x) = f [g(x)] = f [log_b x]

= b^{logb x} = x, x > 0 /

(g ◦ f )(x) = g[f (x)] = g[b^x]

= log_b b^x = x, −∞ < x < ∞

¹

f [g(x)] = x, x ∈ g íì2域 /

g[f (x)] = x, x ∈ f íì2域

1˚ _f D _g Ñ¥ƒb (inverse function), àÇý_. 6ÿuz_, NbƒbDúbƒbÑ¥ƒb_, «u逆 í_, ª彼¤J銷_.

AÍNbDúbí逆4jZkjÖNbDúbíj˙_, à

W _4.

(a) 2e^x+2 − 7 = 5

(b) 5 ln x + 3 = 0

(c) 3 · 2^−0.2t − 4 = 6

(d) 50

1 + 40e^0.2t = 40

<j_{> (a)} ácÜ_, )

e^x+2 = 7 + 5

2 = 6

s邊¦ _ln 1根WNú逆4_,

ln e^x+2 = x + 2 = ln 6

¹

x = ln 6 − 2 ≈ −0.21

(b) ácÜ_, )

ln x = −3 5 s邊¦ _e 1根WNú逆4_,

e^{ln x} = e^−3/5

¹

x = e^−3/5 ≈ 0.55

(c) ácÜ_, )

2^−0.2t = 10 3 s邊¦ _ln 1“_, )

−0.2t ln 2 = ln 10 3

]

t = −5 ln ¹⁰₃

ln 2 = −5(ln 10 − ln 3)

ln 2 ≈ −8.68 (d) ácÜ_, )

1 + 40e^0.2t = 5 4

¹

e^0.2t = 1 40

5

4 − 1



= 1

160 s邊¦ _ln 1“_, )

0.2t = − ln 160 ]

t = −5 ln 160 ≈ −25.38

W _5.

f (x) = a + b ln x /˛ø

f (1) = 2 D _{f (2) = 4}

t° _{f .}

<j_> H _{x = 1,} )

a + b ln 1 = 2

¹

a + b(0) = 2 ] _{a = 2.} yH _{x = 2} D _{a = 2,} )

2 + b ln 2 = 4

¹

b = 4 − 2

ln 2 = 2 ln 2 Ĥ_,

f (x) = 2 + 2 ln 2x