对数函数 TBD¶

\[ \begin{aligned} \log_axy&=\log_ax+\log_ay&\Leftrightarrow\,&a^xa^y=a^{x+y}\\ \log_a\frac{x}{y}&=\log_ax-\log_ay&\Leftrightarrow\,&\frac{a^x}{a^y}=a^{x-y}\\ \log_ax^y&=y\log_ax&\Leftrightarrow\,&(a^x)^y=a^{xy}\\ \log_a\sqrt[y]x&=\frac{\log_ax}y&\Leftrightarrow\,&\sqrt[y]x=x^\frac{1}{y} \end{aligned} \]

\[ \begin{aligned} a^{\log_ax}&=x\\ \log_aa^x&=x \end{aligned} \]

\[ \begin{aligned} \log_ax&=\frac{\log_bx}{\log_ba}\\ \log_ax&=\frac{1}{\log_xa}\\ \log_{a^n}b&=\frac{\log_ab}{n} \end{aligned} \]

\[ \begin{aligned} x^{\log_ay}&=y^{\log_ax}\\ \log_ab\log_bx&=\log_ax \end{aligned} \]

\[ \begin{aligned} \log_a1&=0\\ \log_aa&=a\\ \end{aligned} \]

\[ \begin{aligned} \log_ex&=\ln x\\ \log_2x&=\operatorname{lb}x\\ \log_{10}x&=\lg x \end{aligned} \]