• $$\large x=n+0.{d_1}{d_2}{d_3}\ldots\tag{1}$$

• (1)的表意意味着对于所有的正整数k:

$$\large n+\frac{d_1}{10}+\frac{d_2}{10^2}+\ldots+\frac{d_k}{10^k}\leq x < n+\frac{d_1}{10}+\frac{d_2}{10^2}+\ldots+\frac{d_k}{10^k}+\frac{1}{10^k}\tag{2}$$

• $$\large |z|=\sqrt{x^2+y^2}\tag{3}$$

• $$\large {b^{x+y}}=b^xb^y,(b^x)^y=b^{xy}\tag{5}$$

• $$\large {b^{p/q}}=\sqrt[q]{b^p}\tag{6}$$

• 对于所有实数值$x$，我们定义$b^x$。首先假设$b>1$;如果$x$由式(1)给出，我们要求

$$\Large b^{n+d_1/10+\ldots+d_k/10^k}\leq b^x < b^{n+d_1/10+\ldots+d_k/10^k+1/10^k}\tag{7}$$

• $$\large x=b^{\log_b x}=\log_b(b^x)\tag{9}$$

• 如果$x>0,y>0$

$$\large\log_b(xy)=\log_b x+log_b y\tag{11}$$

• 且,如果$c>0$

$$\large log_b(c^y)=y \log_b c \tag{12}$$

• $$\large lg\;x=\log_2 x \tag{13}$$

• $$\large log_c x = \frac{\log_b x}{\log_b c} \tag{14}$$

• $$\large ln\;x = \log_e x\tag{15}$$