估計 1234 ^ 3235

不用 log，下列何者為1234^3235的最佳估值(解釋)
A.10^4000
B.10^5000
C.10^10000
D.10^20000Sol
1234^3235
=1.234^3235*10^9705
=(1.234^5)^647*10^9705
>2.8613816^647*10^9705
=2.8613816^640*2.8613816^7*10^9705
>(2.8613816^16)^40*1570.4728*10^9705
>20193538^40*1.5704728*10^9708
=2.0193538^40*1.5704728*10^9988
=(2.0193538^16)^2*(2.0193538^8)*1.5704728*10^9988
>76453.698^2*276.50262*1.5704728*10^9988
=7.6453698^2*2.7650262*1.5704728*10^9998
>253.82047*10^9998
=2.5382047*10^10000
So
1234^3235>2.5382047*10^10000
1234^3235
=1.234^3235*10^9705
=(1.234^5)^647*10^9705
<2.862^647*10^9705
=2.862^640*2.862^7*10^9705
<(2.862^16)^40*1573*10^9705
<20270000^40*1.574*10^9708
=2.027^40*1.574*10^9988
=(2.027^16)^2*(2.027^8)*1.574*10^9988
<81250^2*386*1.574*10^9988
=8.125^2*3.86*1.574*10^9998
<410*10^9998
<4.1*10^10000
So
4.1*10^10000>1234^3235.2.5382047*10^10000
(C)

☂雨後陽光☀，條件「不用 log」- log是以10的次方表達，用10的次方不當作用log，是嗎？請示。

1234^3235 ~ 1000^4000 ~ 10^12000 , so the answer should be C or D .

B.10^5000
C.10^10000
D.10^20000

Let 1234^3235 be a

1000^3235 < a < 1600^3235

10^9705 < a < 40^6470

10^9705 < a < ( 10 x 4 ) ^ 6470

10^9705 < a < 10^6470 x 2^12940 ( 2^3 ~ 10 )

10^9705 < a < 10^6470 x 10^4000

10^9705 < a < 10^10470

10^9705 比 10^5000 , 10^20000 更近 10 ^ 10000 ;

10^10470 比 10^5000 , 10^20000 更近 10 ^ 10000 ,

a ~ 10^10000

所以 10 ^ 10000 為最佳估值 ( C )

2010-09-24 14:51:59 補充：
1234^3235 ~ 1000^4000 ~ 10^12000 , 所以答案應是 B , C or D .

2010-09-24 17:55:15 補充：
Exactly ( By using a calculator ) :

1234^3235

= ( 1.234 * 10^3 ) ^ 3235

= 1.234 ^ 3235 * 10^9705

= ( 1.234^5 ) ^ 647 * 10^9705

= 2.861381721051424 ^ 647 * 10^9705

= 2.861381721051424 ^ 642 * 2.861381721051424 ^ 5 * 10^9705

2010-09-24 18:05:10 補充：
~ ( 2.861381721051424 ^ 6 ) ^107 * 191.8134 * 10^9705

~ 548.8514 ^ 107 * 2 * 10^9707

~ 5.488514 ^ 107 * 2 * 10^9921

~ 5.488514 ^ 96 * 5.488514 ^ 11 * 2 * 10^9921

~ [ ( 5.4885 ^ 4 ) ^ 4 ] ^ 6 * 136145264.3968 * 2 * 10^9921

~ ( 907.4332 ^ 4 ) ^ 6 * 2.7729 * 10^9929

~ 678045219748.6893 ^ 6 * 2.7729 * 10^9929

2010-09-24 18:11:09 補充：
~ 6.78045 ^ 6 * 2.7729 * 10^9995

~ 97174.223 * 2.7729 * 10^9995

~ 2.7 * 10^10000

So , the answer is (C) .

By using a calculator ,

1234^3235 ~ 2.5382922309847554511598425907118 * 10^10000

2010-09-24 18:58:29 補充：
2.7 * 10^10000 - 10^10000 < 10^10000 - 10^5000

2.7 * 10^10000 - 10^10000 < 10^20000 -10^10000

So , C. 最佳估值

C.10^10000


1234 ~= 10^3

1234^3235
~= 10^3^3235
~= 10^(3*3235)
~=10^9705
~=10^10000