※ 引述《madduxwin (師出名門)》之銘言：
: 指數不等式問題，請大家指教一下。
: 8^(x-1) + 7*2^(x-2) < = 1 + 7*2^(2x-3)
: 試問：正整數解個數。
8^(x-1)+7*2^(x-2)<=1+7*2(2x-3)
8^(x-1) - 1 <= 7 * [2^(2x-3) - 2^(x-2)]......移項
8^(x-1) - 1 <= 7 * 2^(x-2) * [2^(x-1) - 1].....提出2^(x-2)
8^(x-1) - 1 <= 7/2 * 2^(x-1) * [2^(x-1) - 1]...把2^(x-2)變成2^(x-1) 前面要除以2
另2^(x-1)=a
則上式為
a^3 - 1 <= 7/2 * a * (a-1)
2(a^3) - 7(a^2) + 7a - 2 <=0
2(a^3 - 1) - 7a(a-1) <= 0
2(a-1)(a^2 + a + 1) - 7a(a-1) <= 0
(a-1)(2a^2 + 2a + 2 - 7a) <= 0
(a-1)(2a^2 - 5a + 2) <= 0
(a-1)(2a-1)(a-2) <= 0
- + - +
----------|-------|--------------|---------------數線
1/2 1 2
可得知 a <= 1/2 或 1 <= a <= 2
又a=2^(x-1) 所以a > 0
0 < 2^(x-1) <= 1/2 ,
x <= 0,此時x無正整數解
1 <= 2^(x-1) <= 2,
0 <= x-1 <= 1 , 此時x有兩個正整數解1,2
<Ans> 2個正整數解
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