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4. 指数和对数
1. a x a^x a x log a x \log_a x log a x
乘法群* R + ∗ \mathbf{R}_+^* R + ∗ of 实数 > 0 > 0 > 0 是群同构于加法群 R \mathbf{R} R of 实数 。
引用了 6 · 被以下引用 1 引用了 命题 1, IV § 4 , 定理 2, V § 3 , 推论, V § 1 , 命题 5, V § 1 , IV § 3. 实数的域 , 习题 1, § 4
被以下引用 命题 1, VIII § 1
因为R + ∗ = ] 0 , + ∞ [ \mathbf{R}_+^* = ]0, +\infty[ R + ∗ = ] 0 , + ∞ [ R \mathbf{R} R R \mathbf{R} R 同构的 到 R \mathbf{R} R
从推论到§ 1, no. 3的命题 5，对于每个数a > 0 a > 0 a > 0 R \mathbf{R} R R + ∗ \mathbf{R}_+^* R + ∗ f a f_a f a f a ( 1 ) = a f_a(1) = a f a ( 1 ) = a x ∈ R x \in \mathbf{R} x ∈ R y ∈ R y \in \mathbf{R} y ∈ R
f a ( x + y ) = f a ( x ) f a ( y ) , f a ( − x ) = 1 f a ( x ) ,
f_a(x + y) = f_a(x) f_a(y), \qquad f_a(-x) = \frac{1}{f_a(x)},
f a ( x + y ) = f a ( x ) f a ( y ) , f a ( − x ) = f a ( x ) 1 , 因此，特别是对于所有 n ∈ Z n \in \mathbf{Z} n ∈ Z
f a ( n ) = a n .
f_a(n) = a^n.
f a ( n ) = a n . 因此，对于所有 x ∈ R x \in \mathbf{R} x ∈ R a x a^x a x f a ( x ) f_a(x) f a ( x ) a x a^x a x a > 0 a > 0 a > 0 x ∈ R x \in \mathbf{R} x ∈ R 1 x = 1 1^x = 1 1 x = 1 a ≠ 1 a \neq 1 a = 1 x → a x x \to a^x x → a x R \mathbf{R} R R + ∗ \mathbf{R}_+^* R + ∗
如果R + ∗ \mathbf{R}_+^* R + ∗ a ≠ 1 a \neq 1 a = 1 x → a x x \to a^x x → a x R \mathbf{R} R a a a x ∈ R + ∗ x \in \mathbf{R}_+^* x ∈ R + ∗ log a x \log_a x log a x
(1)
a x + y = a x a y ( x ∈ R , y ∈ R , a > 0 ) ;
a^{x+y} = a^x a^y \qquad (x \in \mathbf{R}, y \in \mathbf{R}, a > 0);
a x + y = a x a y ( x ∈ R , y ∈ R , a > 0 ) ; (2)
a − x = 1 a x ( x ∈ R , a > 0 ) ;
a^{-x} = \frac{1}{a^x} \qquad (x \in \mathbf{R}, a > 0);
a − x = a x 1 ( x ∈ R , a > 0 ) ; (3)
log a 1 = 0 , log a a = 1 ( a > 0 , a ≠ 1 ) ;
\log_a 1 = 0, \quad \log_a a = 1 \qquad (a > 0, a \neq 1);
log a 1 = 0 , log a a = 1 ( a > 0 , a = 1 ) ; (4)
log a ( x y ) = log a x + log a y ( x > 0 , y > 0 ) ;
\log_a (xy) = \log_a x + \log_a y \qquad (x > 0, y > 0);
log a ( x y ) = log a x + log a y ( x > 0 , y > 0 ) ; (5)
log a ( 1 x ) = − log a x ( x > 0 ) ;
\log_a \left( \frac{1}{x} \right) = -\log_a x \qquad (x > 0);
log a ( x 1 ) = − log a x ( x > 0 ) ; (6)
a log a x = x ( x > 0 ) ;
a^{\log_a x} = x \qquad (x > 0);
a l o g a x = x ( x > 0 ) ; (7)
log a a x = x ( x ∈ R ) .
\log_a a^x = x \qquad (x \in \mathbf{R}).
log a a x = x ( x ∈ R ) . 到§ 1, no. 3的命题 5，R \mathbf{R} R R + ∗ \mathbf{R}_+^* R + ∗ y → a x y y \to a^{xy} y → a x y x ∈ R x \in \mathbf{R} x ∈ R y = 1 y = 1 y = 1 a x a^x a x
(8)
( a x ) y = a x y ( x ∈ R , y ∈ R , a > 0 ) ,
(a^x)^y = a^{xy} \qquad (x \in \mathbf{R}, \ y \in \mathbf{R}, \ a > 0),
( a x ) y = a x y ( x ∈ R , y ∈ R , a > 0 ) , 或者，改变记号，
(9)
x y = a y ⋅ log a x ( x > 0 , y ∈ R , a > 0 , a ≠ 1 ) .
x^y = a^{y \cdot \log_a x} \qquad (x > 0, y \in \mathbf{R}, a > 0, a \neq 1).
x y = a y ⋅ l o g a x ( x > 0 , y ∈ R , a > 0 , a = 1 ) . 公式 (8) 表明，对于每个整数 n > 0 n > 0 n > 0 ( a 1 / n ) n = a (a^{1/n})^n = a ( a 1/ n ) n = a a n \sqrt[n]{a} n a a 1 / n a^{1/n} a 1/ n
公式 (7) 和 (9) 表明
(10)
log a ( x y ) = y log a x ( x > 0 , y ∈ R ) ,
\log_a (x^y) = y \log_a x \qquad (x > 0, y \in \mathbf{R}),
log a ( x y ) = y log a x ( x > 0 , y ∈ R ) , 或者，改变记号，
(11)
log a x = log a b ⋅ log b x ( x > 0 , a > 0 , b > 0 , a ≠ 1 , b ≠ 1 ) ,
\log_a x = \log_a b \cdot \log_b x \qquad (x > 0, a > 0, b > 0, a \neq 1, b \neq 1),
log a x = log a b ⋅ log b x ( x > 0 , a > 0 , b > 0 , a = 1 , b = 1 ) , 这是“变换 of 基”的公式。
最后，让我们获得拓扑群R ∗ \mathbf{R}^* R ∗ g g g
log a ( g ( a x ) )
\log_a (g(a^x))
log a ( g ( a x )) 是 R \mathbf{R} R R \mathbf{R} R α ∈ R \alpha \in \mathbf{R} α ∈ R log a ( g ( a x ) ) = α x \log_a (g(a^x)) = \alpha x log a ( g ( a x )) = α x x ∈ R x \in \mathbf{R} x ∈ R g ( x ) = x α g(x) = x^\alpha g ( x ) = x α x > 0 x > 0 x > 0
(12)
( x y ) α = x α y α for all x > 0 , y > 0 and α ∈ R .
(xy)^\alpha = x^\alpha y^\alpha \text{ for all } x > 0, y > 0 \text{ and } \alpha \in \mathbf{R}.
( x y ) α = x α y α for all x > 0 , y > 0 and α ∈ R . 由于公式 (4)，其中约化每隔乘法到加法（唯一与习惯的系统很好地适应的运算），对数长期以来一直是数值计算中不可或缺的工具（参见历史注记） 章）。
当用于此目的时，基选择为a = 10 a = 10 a = 10 log 10 x \log_{10} x log 10 x e e e lim x → 1 , x ≠ 1 log e x ( x − 1 ) = 1 \lim_{x \to 1, x \neq 1} \frac{\log_e x}{(x - 1)} = 1 lim x → 1 , x = 1 ( x − 1 ) l o g e x = 1
2. 函数 a x a^x a x log a x \log_a x log a x
通过章IV, § 2, no. 6的定理5，如果a ≠ 1 , x → a x a \neq 1, x \to a^x a = 1 , x → a x R \mathbf{R} R R + ∗ = ] 0 , + ∞ [ \mathbf{R}_+^* = ]0, + \infty[ R + ∗ = ] 0 , + ∞ [ a > 1 a > 1 a > 1 a 1 = a ≥ 1 = a 0 a^1 = a \geq 1 = a^0 a 1 = a ≥ 1 = a 0 a x a^x a x R + ∗ \mathbf{R}_+^* R + ∗ a x a^x a x R \mathbf{R} R
lim x → + ∞ a x = + ∞ ( a > 1 )
\lim_{x \to +\infty} a^x = +\infty \quad (a > 1)
x → + ∞ lim a x = + ∞ ( a > 1 ) 并且，通过(2)，
lim x → − ∞ a x = 0 ( a > 1 ) .
\lim_{x \to -\infty} a^x = 0 \quad (a > 1).
x → − ∞ lim a x = 0 ( a > 1 ) . 另一方面，如果a < 1 a < 1 a < 1 a x a^x a x x x x + ∞ +\infty + ∞ x x x − ∞ -\infty − ∞ + ∞ +\infty + ∞
!显示 y = a^x (a > 1) 和 y = a^x (a < 1) 的图表
图 1.
!图表显示 y = log_a x (a > 1) 和 y = log_a x (a < 1)
图 2.
从这些属性和 (12) 中，我们推断出，如果 0 < a < b 0 < a < b 0 < a < b x > 0 x > 0 x > 0 a x < b x a^x < b^x a x < b x x < 0 x < 0 x < 0 a x > b x a^x > b^x a x > b x x > 0 x > 0 x > 0 ( b a ) x > 1 \left(\frac{b}{a}\right)^x > 1 ( a b ) x > 1 x < 0 x < 0 x < 0 ( b a ) x < 1 \left(\frac{b}{a}\right)^x < 1 ( a b ) x < 1
R + ∗ \mathbf{R}_+^* R + ∗ log a x \log_a x log a x R \mathbf{R} R a x a^x a x a > 1 a > 1 a > 1 log a x \log_a x log a x x x x 0 0 0 − ∞ -\infty − ∞ x x x + ∞ +\infty + ∞ + ∞ +\infty + ∞ a < 1 a < 1 a < 1 log a x \log_a x log a x x x x 0 0 0 + ∞ +\infty + ∞ x x x + ∞ +\infty + ∞ − ∞ -\infty − ∞
函数 a x a^x a x log a x \log_a x log a x R ‾ \overline{\mathbf{R}} R R ‾ \overline{\mathbf{R}} R R ‾ \overline{\mathbf{R}} R [ 0 , + ∞ ] [0, +\infty] [ 0 , + ∞ ] R ‾ \overline{\mathbf{R}} R + ∞ +\infty + ∞ − ∞ -\infty − ∞ 0 0 0 + ∞ +\infty + ∞
More generally, formula (9) shows that the function x y x^y x y R + ∗ × R \mathbf{R}_+^* \times \mathbf{R} R + ∗ × R R ‾ 2 \overline{\mathbf{R}}^2 R 2 ( x , y ) (x, y) ( x , y ) ( a , b ) (a, b) ( a , b ) R ‾ 2 \overline{\mathbf{R}}^2 R 2 R + ∗ × R \mathbf{R}_+^* \times \mathbf{R} R + ∗ × R ( 0 , 0 ) (0, 0) ( 0 , 0 ) ( + ∞ , 0 ) (+\infty, 0) ( + ∞ , 0 ) ( 1 , + ∞ ) (1, +\infty) ( 1 , + ∞ ) ( 1 , − ∞ ) (1, -\infty) ( 1 , − ∞ ) R ‾ 2 \overline{\mathbf{R}}^2 R 2 x y x^y x y
注扩张通过x y x^y x y 0 0 = 1 0^0 = 1 0 0 = 1
注另外，指数的定义允许我们对所有 a > 0 a > 0 a > 0 Z \mathbf{Z} Z R \mathbf{R} R n → a n n \to a^n n → a n a < 0 a < 0 a < 0
3. 数 > 0 > 0 > 0
拓扑群R \mathbf{R} R R + ∗ \mathbf{R}_+^* R + ∗ > 0 > 0 > 0 ( x i ) (x_i) ( x i ) ( log a x i ) (\log_a x_i) ( log a x i ) a a a > 0 > 0 > 0 ≠ 1 \neq 1 = 1
∏ i x i = a ∑ log a x i .
\prod_i x_i = a^{\sum \log_a x_i}.
i ∏ x i = a ∑ l o g a x i . 同样，由有限数 > 0 > 0 > 0 ( 1 + u n ) (1 + u_n) ( 1 + u n ) 收敛 (章 IV, § 7, no. 6) 当且仅当级数收敛，其一般项为 log a ( 1 + u n ) \log_a (1 + u_n) log a ( 1 + u n )
∏ n = 0 ∞ ( 1 + u n ) = a S n = 0 ∞ log a ( 1 + u n ) .
\prod_{n=0}^{\infty} (1 + u_n) = a^{S_{n=0}^{\infty} \log_a (1 + u_n)}.
n = 0 ∏ ∞ ( 1 + u n ) = a S n = 0 ∞ l o g a ( 1 + u n ) . 因此，对实数> 0 > 0 > 0
习题
参见习题为 § 4 。
