·∫∫Ωñ° §≥‘μ»“ μ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 274

°“√ª√–¬ÿ°μå¢Õß ¡°“√‡™‘߇ âπμ—«·ª√‡¥’¬«

1. ª√–‚¬§¿“…“·≈–ª√–‚¬§ —≠≈—°…≥å

°‘®°√√¡∑’Ë 1

1. „Àâπ—°‡√’¬π‡¢’¬π —≠≈—°…≥å·∑π¢âÕ§«“¡μàÕ‰ªπ’È(1) º≈∫«°¢Õß Õ߇∑à“¢Õß®”π«π®”π«πÀπ÷Ëß°—∫ÀⓇ∑à“¢Õß®”π«ππ—Èπ

...................................................................................................................................(2) x ‡æ‘Ë¡¢÷ÈπÕ’° 6

...................................................................................................................................(3) ®”π«π∑’Ë¡“°°«à“ Õ߇∑à“¢Õß x Õ¬Ÿà 5

...................................................................................................................................(4) º≈∫«°¢Õß®”π«π§Ÿà “¡®”π«π‡√’¬ß°—π

...................................................................................................................................(5) x ≈∫¥â«¬ 5 ‡∑à“°—∫ 2

...................................................................................................................................(6) º≈∫«°¢Õß x °—∫ 8 ¡’§à“‡∑à“°—∫ 11

...................................................................................................................................

3

·∫∫Ωñ° §≥‘μ»“ μ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 2 75

(7) ‡»…Àπ÷Ëß à«π “¡¢Õß y ‰¡à‡∑à“°—∫ 9...................................................................................................................................

(8) m ≈∫¥â«¬ 3 πâÕ¬°«à“ 10

...................................................................................................................................(9) º≈§Ÿ≥¢Õß 7 ·≈– n ¡“°°«à“ 21

...................................................................................................................................(10) “¡‡∑à“¢Õߺ≈∫«°¢Õß®”π«π®”π«πÀπ÷Ëß°—∫ 5 ‰¡à‡°‘π 20

...................................................................................................................................(11) º≈∫«° Õ߇∑à“¢Õß®”π«π®”π«πÀπ÷Ëß°—∫ 5 ‰¡àμË”°«à“ 15

...................................................................................................................................(12) º≈∫«°¢Õß®”π«ππ—∫ “¡®”π«π‡√’¬ß°—π‡∑à“°—∫ 21

...................................................................................................................................(13) ‡»… “¡ à«π ’Ë¢Õß®”π«π®”π«πÀπ÷Ëß≈∫¥â«¬‡»…Àπ÷Ëß à«π ’Ë¢Õß®”π«ππ—Èπ‡∑à“°—∫ 10

...................................................................................................................................(14) °”≈—ß ÕߢÕß®”π«π®”π«πÀπ÷Ëß≈∫¥â«¬ 5 ‡∑à“°—∫ 20

...................................................................................................................................(15) ’ˇ∑à“¢Õß®”π«π®”π«πÀπ÷Ëß∫«°¥â«¬‡»…Àπ÷Ëß à«π ’Ë¢Õß®”π«ππ—Èπ‡∑à“°—∫ 34

...................................................................................................................................2. „Àâπ—°‡√’¬π‡ª≈’ˬπª√–‚¬§μàÕ‰ªπ’ȇªìπª√–‚¬§¿“…“

(1) x2 = 3...................................................................................................................................

(2) x5 = 5...................................................................................................................................

(3) 34 = 12

...................................................................................................................................(4)

153

= 5

...................................................................................................................................

·∫∫Ωñ° §≥‘μ»“ μ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 276

(5) 7x = 35

...................................................................................................................................(6)

x15

= 3...................................................................................................................................

(7) 2x3x = 25

...................................................................................................................................(8) 5(x2) = 30

...................................................................................................................................(9) x

26 = 30

...................................................................................................................................(10)

15

x1

10x = 10

...................................................................................................................................3. „Àâπ—°‡√’¬πæ‘®“√≥“«à“¢âÕ§«“¡„¥‡ªìπ ¡°“√·≈–¢âÕ§«“¡„¥‡ªìπÕ ¡°“√

¡°“√ Õ ¡°“√ ‰¡à‡ªìπ∑—Èß ÕßÕ¬à“ß(1) 175 = 57 ..................... ..................... .......................................

(2) 2x 10 ..................... ..................... .......................................

(3) 3x2x ..................... ..................... .......................................

(4) 5c2 1 ..................... ..................... .......................................

(5) 4m = 0 ..................... ..................... .......................................

(6) 4x3y ..................... ..................... .......................................

(7) 9h = 27 ..................... ..................... .......................................

(8) 4x5 = 19 ..................... ..................... .......................................

(9) 3x2 = x8 ..................... ..................... .......................................

(10) 5x2x = 6 ..................... ..................... .......................................

(11) 5x π 100 ..................... ..................... .......................................

(12) 2x7 = x11 ..................... ..................... .......................................

(13)

x x3 2

= 10 ..................... ..................... .......................................

(14) x5 x7 ..................... ..................... .......................................

(15) 9x6(34x) π 207 ..................... ..................... .......................................

·∫∫Ωñ° §≥‘μ»“ μ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 2 77

4. „Àâπ—°‡√’¬πÀ“®”π«π¡“·∑πμ—«·ª√‡æ◊ËÕ∑”„Àâª√–‚¬™πåμàÕ‰ªπ’ȇªìπ®√‘ß(1) 3x = 12 ..............................................................................................(2) 4x = 8 ..............................................................................................(3) 5x = 4x2 ..............................................................................................(4) 6x3x = 9 ..............................................................................................(5) 7x2x = 15 ..............................................................................................(6) 2x5 = x1 ..............................................................................................(7) 3x4 = 2x3 ..............................................................................................(8) x15 = 8 ..............................................................................................(9)

15

x = 5 ..............................................................................................

(10) x3(x2) = 2x2 ..............................................................................................(11) 3x4x = 1 ..............................................................................................(12)

17

x = 3 ..............................................................................................

2. §”μÕ∫¢Õß ¡°“√

°‘®°√√¡∑’Ë 2

1. „Àâπ—°‡√’¬π‡¢’¬πª√–‚¬§ —≠≈—°…≥å·≈â«À“§”μÕ∫¢Õß ¡°“√(1) ®”π«π®”π«πÀπ÷Ëß≈∫¥â«¬ 2 ‡∑à“°—∫ 3

.....................................................................................................................................(2) °”≈—ß ÕߢÕß®”π«π®”π«πÀπ÷Ëß≈∫¥â«¬ 6 ‡∑à“°—∫ 30

.....................................................................................................................................(3) ®”π«π®”π«πÀπ÷ËßÀ“√¥â«¬ 15 ‡∑à“°—∫ 3

.....................................................................................................................................

·∫∫Ωñ° §≥‘μ»“ μ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 278

(4) ÀⓇ∑à“¢Õߺ≈∫«°¢Õß®”π«π®”π«πÀπ÷Ëß°—∫ 2 ‡∑à“°—∫ 30

.....................................................................................................................................(5) ‡»…Àπ÷Ëß à«πÀâ“¢Õß®”π«π®”π«πÀπ÷Ëß≈∫¥â«¬‡»…Àπ÷Ëß à«π ‘∫¢Õß®”π«ππ—Èπ‡∑à“°—∫

10

.....................................................................................................................................

2. „Àâπ—°‡√’¬πμ√«® Õ∫¥Ÿ«à“®”π«π∑’Ë°”Àπ¥„Àâ„π«ß‡≈Á∫ [ ] ‡ªìπ§”μÕ∫¢Õß ¡°“√À√◊Õ‰¡à

(1) 2x5 = 1 [3] .....................................................................................

(2) 3x7 = 11 [6] .....................................................................................

(3) 4x40 = 80 [10] .....................................................................................

(4) 5x9 = 4x10 [1] .....................................................................................

(5) 6x4 = 2x8 [3] .....................................................................................

(6) 7p = 213p [7] .....................................................................................

(7) 8q3q = 6q10 [10] .....................................................................................

(8) 3b = 5b18 [9] .....................................................................................

(9) 2t = 3t11 [11] .....................................................................................

(10) 9x = 7x18 [9] .....................................................................................

3. „Àâπ—°‡√’¬πÀ“§”μÕ∫¢Õß ¡°“√μàÕ‰ªπ’È ‚¥¬«‘∏’·∑π§à“

(1) 3x = 12 ........................ (2) 4x = 20 .............................

(3) 5x = 30 ........................ (4)x3 = 12 .............................

(5)x4 = 20 ........................ (6)

x5 = 30 .............................

(7)23a

= 3 ........................ (8)a3

2 = 1 .............................

(9)a9 = 1

3........................ (10)

b7 = 1

3.............................

(11)25c = 4

5........................ (12)

35d = 2 1

5.............................

·∫∫Ωñ° §≥‘μ»“ μ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 2 79

(13)r3

2 = 4 ........................ (14) pp

2 = 1 .............................

(15) x2 = 9 ........................ (16) x

3 = 1,000 .............................

(17)x

2

4 = 9 ........................ (18) x7 = 3x7 .............................

(19) 0.5y3 = 0.3y7 ........................ (20) x13

x = 4 .............................

(21) x15

x = 4 ........................ (22) y14

y = 34

y2 .............................

3. °“√·°â ¡°“√

°‘®°√√¡∑’Ë 3

1. „Àâπ—°‡√’¬π„™â ¡∫—μ‘ ¡¡“μ√‡μ‘¡ª√–‚¬§„Àâ ¡∫Ÿ√≥å(1) „Àâ 2x = 8 ¥—ßπ—Èπ 8 = .....................................................(2) „Àâ 3(xy) = 3x3y ¥—ßπ—Èπ 3x3y = .....................................................(3) „Àâ ab = ba ¥—ßπ—Èπ ba = .....................................................(4) „Àâ (23)x = 2x3x ¥—ßπ—Èπ 2x3x = .....................................................(5) „Àâ 5x10 = 5(x2) ¥—ßπ—Èπ 5(x2) = .....................................................

2. „Àâπ—°‡√’¬π„™â ¡∫—μ‘°“√∂à“¬∑Õ¥‡μ‘¡ª√–‚¬§„Àâ ¡∫Ÿ√≥å(1) ∂â“ a = b ·≈– b = c ·≈â« .............................................................................................(2) ∂â“ 2xx = (21)x ·≈– (21)x = x ·≈â« ....................................................................(3) ∂â“ 2xx = (21)x ·≈– (21)x = 3x ·≈â«.......................................................(4) ∂â“ 2x = (11)x ·≈– (11)x = xx ·≈â« ........................................................(5) ∂â“ 3x5x = (35)x ·≈– (35)x = 8x ·≈â« .................................................................

·∫∫Ωñ° §≥‘μ»“ μ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 280

3. „Àâπ—°‡√’¬π„™â ¡∫—μ‘°“√∫«° ¡∫—μ‘°“√§Ÿ≥ ·≈– ¡∫—μ‘°“√·®°·®ß‡μ‘¡ª√–‚¬§„Àâ ¡∫Ÿ√≥å(1) ∂â“ x5 = 9 ·≈â« x5(5) = ....................................................................................

(2) ∂â“ x2

3 = 5 ·≈â« x2

3(3) = ..................................................................................

(3) ∂â“ 5x = 4 ·≈â« 15

5x = ...............................................................................................

(4) ∂â“ 2(x2) = 2x4 ·≈â« 3(x3) = ..................................................................................

(5) ∂â“ 25

x = 10 ·≈â«

52

25

x = .....................................................................................

4. „Àâπ—°‡√’¬π„™â ¡∫—μ‘°“√∫«° ¡∫—μ‘°“√§Ÿ≥ ·≈– ¡∫—μ‘°“√·®°·®ß‡μ‘¡ª√–‚¬§„Àâ ¡∫Ÿ√≥å

(1) 2x5 = 25

........................................................... ¡∫—μ‘°“√∫«°

2x = 20

........................................................... ¡∫—μ‘°“√§Ÿ≥

x = 10

(2) 35

x4 = 8

........................................................... ¡∫—μ‘°“√∫«°35

x = 12

........................................................... ¡∫—μ‘°“√§Ÿ≥

x = 20

(3) 3(2x5) = 27

........................................................... ¡∫—μ‘°“√·®°·®ß

6x15 = 27

........................................................... ¡∫—μ‘°“√∫«°

6x = 12

........................................................... ¡∫—μ‘°“√§Ÿ≥

x = 2

·∫∫Ωñ° §≥‘μ»“ μ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 2 81

(4) 33 = 3(2x5)

........................................................... ¡∫—μ‘°“√§Ÿ≥

11 = 2x5

........................................................... ¡∫—μ‘°“√∫«°

6 = 2x

........................................................... ¡∫—μ‘°“√§Ÿ≥

3 = x

5. „Àâπ—°‡√’¬π·°â ¡°“√μàÕ‰ªπ’Èæ√âÕ¡μ√«® Õ∫§”μÕ∫(1) 2(x1) = 10

«‘∏’∑” 2(x1) = 10

x1 =102

......................................................................................................................

......................................................................................................................

......................................................................................................................μ√«® Õ∫§”μÕ∫ 2(x1) = 10

............................................................................................... .

............................................................................................... .

............................................................................................... .¥—ßπ—È𠧔μÕ∫¢Õß ¡°“√§◊Õ 4

(2) 3(x1) = 6

«‘∏’∑” 3(x1) = 6

x1 =

63

......................................................................................................................

......................................................................................................................

......................................................................................................................

·∫∫Ωñ° §≥‘μ»“ μ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 282

μ√«® Õ∫§”μÕ∫ 3(x1) = 6

............................................................................................... .

............................................................................................... .

............................................................................................... .¥—ßπ—È𠧔μÕ∫¢Õß ¡°“√§◊Õ 1

(3) 7x11 = 6(x1)

«‘∏’∑” 7x11 = 6(x1)

7x11 = 6x6

......................................................................................................................

......................................................................................................................μ√«® Õ∫§”μÕ∫ 7x11 = 6(x1)

............................................................................................... .

............................................................................................... .

............................................................................................... .¥—ßπ—È𠧔μÕ∫¢Õß ¡°“√§◊Õ 5

(4) 5(x2) = 3(x6)

«‘∏’∑” 5(x2) = 3(x6)

5x10 = 3x18

......................................................................................................................

......................................................................................................................

......................................................................................................................μ√«® Õ∫§”μÕ∫ 5(x2) = 3(x6)

............................................................................................... .

............................................................................................... .

............................................................................................... .¥—ßπ—È𠧔μÕ∫¢Õß ¡°“√§◊Õ 14

·∫∫Ωñ° §≥‘μ»“ μ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 2 83

(5) 0.2(x1)0.3(x2) = 1.2

«‘∏’∑” 0.2(x1)0.3(x2) = 1.2

......................................................................................................................

......................................................................................................................

......................................................................................................................

......................................................................................................................

......................................................................................................................

......................................................................................................................μ√«® Õ∫§”μÕ∫ 0.2(x1)0.3(x2) = 1.2

............................................................................................... .

............................................................................................... .

............................................................................................... .

................................................................................................¥—ßπ—È𠧔μÕ∫¢Õß ¡°“√§◊Õ 4

(6)25

x5 = 11

«‘∏’∑” ....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

μ√«® Õ∫§”μÕ∫ ............................................................................................................................................................................................... ................................................................................................ ................................................................................................ .................................................................................................

......................................................................................................................

·∫∫Ωñ° §≥‘μ»“ μ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 284

(7) 2x2 =

16 9

x

«‘∏’∑” ......................................................................................................................

......................................................................................................................

......................................................................................................................

......................................................................................................................

......................................................................................................................

......................................................................................................................

μ√«® Õ∫§”μÕ∫ ................................................................................................

............................................................................................... .

................................................................................................

......................................................................................................................

(8)

xx

34

=

3 54

x

«‘∏’∑” ......................................................................................................................

......................................................................................................................

......................................................................................................................

......................................................................................................................

......................................................................................................................

......................................................................................................................

......................................................................................................................

μ√«® Õ∫§”μÕ∫ ................................................................................................

............................................................................................... .

............................................................................................... .

................................................................................................

......................................................................................................................

·∫∫Ωñ° §≥‘μ»“ μ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 2 85

(9)

xx

113 = 3

2, x π 3

«‘∏’∑” ......................................................................................................................

......................................................................................................................

......................................................................................................................

......................................................................................................................

......................................................................................................................

μ√«® Õ∫§”μÕ∫ ................................................................................................

............................................................................................... .

............................................................................................... .

................................................................................................

......................................................................................................................

(10) 7(x4a) = 14(x3a)

«‘∏’∑” ......................................................................................................................

......................................................................................................................

......................................................................................................................

......................................................................................................................

......................................................................................................................

μ√«® Õ∫§”μÕ∫ ................................................................................................

............................................................................................... .

............................................................................................... .

......................................................................................................................

(11)34

(3x1) = 87

(4x8)11

«‘∏’∑” ......................................................................................................................

......................................................................................................................

·∫∫Ωñ° §≥‘μ»“ μ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 286

......................................................................................................................

......................................................................................................................

......................................................................................................................

......................................................................................................................

......................................................................................................................

μ√«® Õ∫§”μÕ∫ 34

(3x1) =87

(4x8)11

............................................................................................... .

............................................................................................... .

............................................................................................... .

................................................................................................

......................................................................................................................

(12)

2 4

3

3 1

2

( )

( )x x

= 1

2 1

3

( )x

«‘∏’∑” ......................................................................................................................

......................................................................................................................

......................................................................................................................

......................................................................................................................

......................................................................................................................

......................................................................................................................

...................................................................................................................................

...................................................................................................................................

...................................................................................................................................

...................................................................................................................................

......................................................................................................................

·∫∫Ωñ° §≥‘μ»“ μ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 2 87

(13)

12 1x

=

11x

«‘∏’∑” ......................................................................................................................

......................................................................................................................

......................................................................................................................

......................................................................................................................

μ√«® Õ∫§”μÕ∫ ................................................................................................

............................................................................................... .

............................................................................................... .

......................................................................................................................

6. „Àâπ—°‡√’¬πÀ“§à“¢Õß x ®“° ¡°“√μàÕ‰ªπ’È

(1) 25% ¢Õß x = 40

«‘∏’∑” 25% ¢Õß x = 40

25100

x = 40

x = ..........................................................................................

........................................................................................................................

(2) x15% ¢Õß x = 3.45

«‘∏’∑” x15100

x= 3.45

x0.15x = .................................................................................

........................................................................................................................

........................................................................................................................

........................................................................................................................

·∫∫Ωñ° §≥‘μ»“ μ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 288

(3) x5 = 33313 % ¢Õß x

«‘∏’∑” x5 = 3 100

3100

x

........................................................................................................................

........................................................................................................................

........................................................................................................................

........................................................................................................................

........................................................................................................................

7. °”Àπ¥ A = k(axb) ®ßÀ“ x ‡¡◊ËÕ A = 10, k = 5, a = 1, b = 3

«‘∏’∑” A = k(axb)

Ak

= axb

ax =Ak

b

x =1a

Ak

b ÊËÁ

ˆ¯

x = ..........................................................................................

........................................................................................................................

........................................................................................................................

·∫∫Ωñ° §≥‘μ»“ μ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 2 89

4. ‚®∑¬å ¡°“√

°‘®°√√¡∑’Ë 41. °”Àπ¥μ—«·ª√ x „Àâπ—°‡√’¬π‡¢’¬π —≠≈—°…≥宓°¢âÕ§«“¡μàÕ‰ªπ’È

(1) º≈∫«°¢Õß®”π«π®”π«πÀπ÷Ëß°—∫ 5

.....................................................................................................................................

(2) ®”π«π´÷Ëß¡“°°«à“®”π«π®”π«πÀπ÷ËßÕ¬Ÿà 7

.....................................................................................................................................

(3) ®”π«π´÷ËßπâÕ¬°«à“®”π«π®”π«πÀπ÷ËßÕ¬Ÿà 5

.....................................................................................................................................

(4) º≈§Ÿ≥¢Õß 3 °—∫®”π«π®”π«πÀπ÷Ëß

.....................................................................................................................................

(5) ÀⓇ∑à“¢Õß®”π«π®”π«πÀπ÷Ëß

.....................................................................................................................................

(6) ·∫àß®”π«π®”π«πÀπ÷ËßÕÕ°‡ªìπ 5 à«π‡∑à“Ê °—π

.....................................................................................................................................

(7) 5 ¡“°°«à“®”π«π®”π«πÀπ÷Ëß

.....................................................................................................................................

(8) 35 πâÕ¬°«à“ÀⓇ∑à“¢Õß®”π«π®”π«πÀπ÷Ëß

.....................................................................................................................................

2. „Àâπ—°‡√’¬π‡¢’¬π —≠≈—°…≥å·≈– ¡°“√®“°¢âÕ§«“¡μàÕ‰ªπ’È

(1) Õ߇∑à“¢Õß®”π«π®”π«πÀπ÷Ëß¡“°°«à“ 3 Õ¬Ÿà 7

.....................................................................................................................................

·∫∫Ωñ° §≥‘μ»“ μ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 290

(2) ÀⓇ∑à“¢Õߺ≈∫«°¢Õß®”π«π®”π«πÀπ÷Ëß°—∫ 5 ‡ªìπ 20

.....................................................................................................................................(3) ®”π«π®”π«πÀπ÷Ë߬°°”≈—ß Õ߇∑à“°—∫ 49

.....................................................................................................................................(4) º≈∫«°¢Õß®”π«π‡μÁ¡∫«° “¡®”π«π‡√’¬ß°—π‡ªìπ 63

.....................................................................................................................................(5) π“¡À≠â“√Ÿª ’ˇÀ≈’ˬ¡º◊πºâ“·ÀàßÀπ÷Ëß¡’¥â“𬓫¬“«°«à“¥â“π°«â“ß 5 ‡¡μ√ ·≈–

π“¡À≠â“π’Ȭ“« 25 ‡¡μ√.....................................................................................................................................

(6) æàÕπ”‡ß‘π®”π«πÀπ÷Ëß√«¡°—∫‡ß‘π 1,000 ∫“∑ ·≈â«·∫àß„Àâ≈Ÿ° 2 §π ‰¥â√—∫§π≈– 300

∫“∑.....................................................................................................................................

(7) √“¬‰¥â¢Õß™“¬§πÀπ÷Ë߇æ‘Ë¡¢÷Èπ 5% ‡¢“®–‰¥â‡ß‘π‡æ‘Ë¡¢÷Èπ 500 ∫“∑.....................................................................................................................................

(8) º≈∫«°¢Õß¢π“¥¢Õß¡ÿ¡¿“¬„π¢Õß√Ÿª “¡‡À≈’ˬ¡√ŸªÀπ÷Ëß ¡ÿ¡·√°¡’¢π“¥‡ªìπ Õ߇∑à“¢Õß¡ÿ¡∑’Ë Õß ·≈–¡ÿ¡∑’Ë “¡¡’¢π“¥‡ªìπ “¡‡∑à“¢Õß¡ÿ¡∑’Ë Õß.....................................................................................................................................

3. º≈∫«°¢Õß®”π«π§Ÿà 3 ®”π«π‡√’¬ß°—π‡ªìπ 114 ®ßÀ“®”π«π∑’ËπâÕ¬∑’Ë ÿ¥«‘∏’∑” „Àâ®”π«π§Ÿà∑’ËπâÕ¬∑’Ë ÿ¥‡ªìπ x

®”π«π§ŸàÕ’° Õß®”π«π∂—¥‰ª§◊Õ x2 ·≈– ...........................................................‚®∑¬å°”Àπ¥º≈∫«°¢Õß®”π«π§Ÿà∑—Èß “¡®”π«π‡∑à“°—∫ ........................................ ¡°“√§◊Õ ...........................................................................................................

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...........................................................................................................¥—ßπ—Èπ ®”π«π∑’ËπâÕ¬∑’Ë ÿ¥§◊Õ 36

·∫∫Ωñ° §≥‘μ»“ μ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 2 91

4. ®”π«π‡μÁ¡∫«° 5 ®”π«π‡√’¬ß°—π√«¡°—π‰¥â 120 ®ßÀ“º≈∫«°¢Õß®”π«π∑’Ë¡“°∑’Ë ÿ¥°—∫®”π«π∑’ËπâÕ¬∑’Ë ÿ¥«‘∏’∑” „Àâ®”π«π‡μÁ¡∫«°®”π«π∑’Ë “¡ (®”π«π°≈“ß) ‡ªìπ x

®”π«π‡μÁ¡∫«° 5 ®”π«π‡√’¬ß°—π §◊Õ ...................................................................®”π«π∑—ÈßÀ¡¥√«¡°—π‰¥â ....................................................................................... ¡°“√§◊Õ ...........................................................................................................

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...........................................................................................................º≈∫«°¢Õß®”π«π¡“°∑’Ë ÿ¥°—∫®”π«ππâÕ¬∑’Ë ÿ¥§◊Õ.............................................................................................................................................................................................................................................................................................................................................................................................

5. ®”π«π®”π«πÀπ÷Ëß¡’ ÕßÀ≈—° ´÷Ëßμ—«‡≈¢„πÀ≈—°Àπ૬πâÕ¬°«à“μ—«‡≈¢„πÀ≈—° ‘∫Õ¬Ÿà 4 ·μà

º≈μà“ߢÕßμ—«‡≈¢∑—Èß Õ߇ªìπ 121

¢Õß®”π«ππ—Èπ ®ßÀ“®”π«ππ—Èπ

«‘∏’∑” „ÀâÀ≈—° ‘∫‡ªìπ xÀ≈—°Àπ૬‡ªìπ .......................................................................................................®”π«ππ—Èπ§◊Õ .........................................................................................................

‚®∑¬å°”Àπ¥ º≈μà“ߢÕßμ—«‡≈¢∑—Èß Õ߇ªìπ 121

¢Õß®”π«ππ—Èπ

¡°“√§◊Õ .......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

μ—«‡≈¢„πÀ≈—° ‘∫§◊Õ ...............................................................................................μ—«‡≈¢„πÀ≈—°Àπ૬§◊Õ ..........................................................................................¥—ßπ—Èπ ®”π«ππ—Èπ§◊Õ ..............................................................................................

·∫∫Ωñ° §≥‘μ»“ μ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 292

6. ®”π«π Õß®”π«π√«¡°—π‰¥â 105 ∂â“π”®”π«ππâÕ¬‰ªÀ“√®”π«π¡“° ®–‰¥âº≈≈—æ∏å 7 ‡À≈◊Õ‡»… 1 ®ßÀ“®”π«π¡“°«‘∏’∑” „Àâ®”π«π¡“°‡ªìπ x

®”π«ππâÕ¬§◊Õ .......................................................................................................π”®”π«ππâÕ¬‰ªÀ“√®”π«π¡“°®–‰¥âº≈≈—æ∏å 7 ‡À≈◊Õ‡»… 1 ¡°“√§◊Õ ...........................................................................................................

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7. §«“¡¬“«¢Õߥâ“π∑—Èß “¡¢Õß√Ÿª “¡‡À≈’ˬ¡‡ªìπ x, x7, x8 ‡´π쑇¡μ√ μ“¡≈”¥—∫ ·≈–‡ âπ√Õ∫√Ÿª¬“« 30 ‡´π쑇¡μ√ ®ßÀ“§«“¡¬“«¢Õߥâ“π∑—Èß “¡¢Õß√Ÿª “¡‡À≈’ˬ¡«‘∏’∑”

„Àâ¥â“π∑—Èß “¡¢Õß√Ÿª “¡‡À≈’ˬ¡¬“« ................................................... ‡´π쑇¡μ√‡ âπ√Õ∫√Ÿª¬“« ..................................................................................... ‡´π쑇¡μ√‚®∑¬å°”À𥇠âπ√Õ∫√Ÿª¬“« ................................................................ ‡´π쑇¡μ√ ¡°“√§◊Õ ...........................................................................................................

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...........................................................................................................¥—ßπ—È𠧫“¡¬“«¢Õߥâ“π∑—Èß “¡¢Õß√Ÿª “¡‡À≈’ˬ¡‡ªìπ .................................................................................................................................... ‡´π쑇¡μ√ μ“¡≈”¥—∫

xx 7

x 8

·∫∫Ωñ° §≥‘μ»“ μ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 2 93

8. ¥â“π∑’Ë ÕߢÕß√Ÿª “¡‡À≈’ˬ¡√ŸªÀπ÷Ëß —Èπ°«à“ 2 ‡∑à“¢Õߥâ“π∑’ËÀπ÷Ëß 5 ‡´π쑇¡μ√ ·≈–¥â“π∑’Ë “¡¬“«°«à“¥â“π∑’ËÀπ÷Ëß 11 ‡´π쑇¡μ√ ∂ⓧ«“¡¬“«‡ âπ√Õ∫√Ÿª 54 ‡´π쑇¡μ√ ®ßÀ“§«“¡¬“«·μà≈–¥â“π«‘∏’∑”

„Àâ¥â“π∑—Èß “¡¢Õß√Ÿª “¡‡À≈’ˬ¡¬“« ................................................... ‡´π쑇¡μ√‡ âπ√Õ∫√Ÿª¬“« ..................................................................................... ‡´π쑇¡μ√‚®∑¬å°”À𥇠âπ√Õ∫√Ÿª¬“« ................................................................ ‡´π쑇¡μ√ ¡°“√§◊Õ ...........................................................................................................

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...........................................................................................................¥—ßπ—È𠧫“¡¬“«¢Õߥâ“π∑—Èß “¡¢Õß√Ÿª “¡‡À≈’ˬ¡‡ªìπ .................................................................................................................................... ‡´π쑇¡μ√ μ“¡≈”¥—∫

xx 11

2x 5

·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 294

1. ∂â“ 2x5 = 38x ·≈â« 111

x1 ¡’§à“‡∑à“‰√

1. 1 2. 2

3. 3 4. 5

2. ∂â“ 23

x13

x = 15 ·≈â« x225 ¡’§à“‡∑à“‰√

1. 95 2. 100

3. 200 4. 225

3. ∂â“ 7.50.3x = 2.50.7x ·≈â« 15

x1 ¡’

§à“‡∑à“‰√1. 0 2. 1

3. 2 4. 3

4. ∂â“ 243x = 4(3x)5x ·≈â« x ¡’§à“‡∑à“‰√1. 1 2. 3

3. 4 4. 12

5. ∂â“ 5(33x)18 = 4(5x3) ·≈â« x ¡’§à“‡∑à“‰√

1. 0 2. 1

3. 2 4. 3

·∫∫∑¥ Õ∫º≈ —¡ƒ∑∏‘Ï∑“ß°“√‡√’¬πª√–®”Àπ૬

µÕπ∑’Ë 1 „Àâπ—°‡√’¬π‡≈◊Õ°§”µÕ∫∑’Ë∂Ÿ°µâÕß

6. ∂â“ 7(2x)4x = 3(2x3) ·≈â« x ¡’§à“‡∑à“‰√

1. 1 2. 2

3. 3 4. 4

7. ∂â“ 3(8x5)5x = x(4x7) ·≈â« x ¡’

§à“‡∑à“‰√

1. 0 2. 1

3. 3 4. 15

8. ∂â“ 11a4a = 12

(26a)3a ·≈â« a ¡’

§à“‡∑à“‰√

1. 5 2. 3

3. 1 4. 2

9. ∂â“ 13b3(3b3) = 17 ·≈â« b ¡’§à“‡∑à“‰√

1. 3 2. 1

3. 2 4. 4

10. ∂â“ 13x3(x5) = 15(1x) ·≈â« x ¡’§à“

‡∑à“‰√1. 1 2. 3

3. 5 4. 6

·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 2 95

µÕπ∑’Ë 2 „Àâπ—°‡√’¬π·°â ¡°“√µàÕ‰ªπ’È

1.

2 15

x = 3x5

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2.

x 63

x =

5 123

x

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3. 7

14

(x4) = 18

(x1)

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·∫∫Ωñ° §≥‘µ»“ µ√åæ◊Èπ∞“π ¡.2 ¿“§‡√’¬π∑’Ë 296

4. π“¡√Ÿª ’ˇÀ≈’ˬ¡º◊πºâ“¡’¥â“𬓫¬“«°«à“¥â“π°«â“ß 3 ‡¡µ√ ∂â“«—¥§«“¡¬“«√Õ∫ π“¡‰¥â

54 ‡¡µ√ ®ßÀ“æ◊Èπ∑’Ë¢Õß π“¡

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5. Àπ—ß ◊Õ‡≈à¡Àπ÷Ëß¡’ 180 Àπâ“ Õà“π‰ª·≈⫇»… Õß à«πÀâ“¢Õß®”π«πÀπâ“∑—ÈßÀ¡¥ ®ßÀ“

®”π«πÀπâ“∑’ˬ—߉¡àÕà“π

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