2011-12-28
1. 什么是对数
2x = 1/4, 3x = 2187, 4x = 256, 5x = 625, 6x = 1/216, 7x = 343, 8x = 32769, 9x = 81, 10x = 1000000. 上面各式中的 x 分别是多少?
当 a > 0,a≠1 时, ax = N ←→ x = logaN ,x 就是以 a 为底的对数(Logarithm),a 叫做对数的底数,N 叫做真数. 以10为底的对数叫做常用对数(Common Logarithm),记做 lgN . 以 e (e = 2.071828 … )为底的对数叫做自然对数(Natural Logarithm),记做 ln N.
- 2x = 1/4 ←→ x = log2(1/4)= -2
- 3x = 2187 ←→ x = log3 2187= 7
- 4x = 256 ←→ x = log4 256= 4
- 5x = 625 ←→ x = log5 625= 4
- 6x = 2187 ←→ x = log6 (1/216)= -3
- 7x = 343 ←→ x = log7 343= 3
- 8x = 32768 ←→ x = log8 32768= 5
- 9x = 81 ←→ x = log9 81= 2
- 10x = 1000000 ←→ x = lg 100000 = 6
- loga 1 = 0
- loga a = 1
2. 对数的运算
当 a > 0,a≠1,M > 0,N > 0 时
- loga (M · N)= loga M + loga N
- loga (M / N)= loga M – loga N
- loga Mn = nloga M (n∈R)
证明: 设 am = M, an = N
∴ am · an= am+n= M · N
又∵ loga M = m, loga N = n
∴ loga (M · N)= m + n = loga M + loga N
3. 对数的换底公式
练习:
1. 计算:
2. 化简:
3. 已知 (lgx)2 – lgx2 – 3 = 0, 求x.
4. 已知lgx – lgy = a, 求 lg5x3 – lg5y3 的值.
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