一道SAT2数学题目：Theproductof45,454,545,454,545and1,234containshowmanydigits?A.14A.14B.15C.16D.17E.18

Question

一道SAT2数学题目：The product of 45,454,545,454,545 and 1,234 contains how many digits?A.14
A.14
B.15
C.16
D.17
E.18

Answer 1

45,454,545,454,545和1234之积有多少位数字?
这里用近似方法和对数做.
45 ≈ 100/2 (实际是45 < 100/2)
1234 ≈ 1024 = 2^10 (实际是1234 > 1024)
454*545 ≈ 500*500 = 1000*1000/(2*2)
log(45*454*545*454*545*1234)
= log45 + 2log(454*545) + log1234
≈ log(100/2) + 2log(1000,000/4) + log2^10
= 2 - log2 + 2(6 - 2log2) + 10log2
= 14 + 5log2
≈ 14 + 5*0.3
= 15.5
积有16位数字
因为原题中逗号后有空格,所以理解为6个数相乘.如敲错了,实际上是两个数相乘,则简单得多.只取二者的前两位:
45*12 = 540
后面还有12 + 2位,共3+12+2 = 17位