HDOJ--ACMSteps--2.1.8--Leftmost Digit-(取对数,数学)

Problem Description
Given a positive integer N, you should output the leftmost digit of N^N.

Input
The input contains several test cases. The first line of the input is a single integer T which is the number of test cases. T test cases follow.
Each test case contains a single positive integer N(1<=N<=1,000,000,000).

Output
For each test case, you should output the leftmost digit of N^N.

Sample Input

2
3
4

Sample Output

2
2

Hint

In the first case, 3 * 3 * 3 = 27, so the leftmost digit is 2.
In the second case, 4 * 4 * 4 * 4 = 256, so the leftmost digit is 2.

Author
Ignatius.L

首先,暴力算是一定会超时的,而且这么大的数也存不下

这道题运用了取对数来缩小运算范围,

推导过程如下:

设M=N^N,则log10(M)=Nlog10(N);

所以M=10^(Nlog10(N));

继续转化

令N=x10^y 例如:155555555=1.5555555510^8;

M=10^{(N*log10(x*10}y)=10^{(N*(y+log10x))=10}(Ny)log10x

so, 只要求出log10x即可

取整即可