# 对一般的对数函数求导的时候，通常可以先转为自然对数

## 二、解析

$$\textcolor{yellow}{ \left( \log_{a} ^{x} \right) ^{\prime} = \frac{1}{x \ln a} }$$

$$\textcolor{yellow}{ \left( \ln x \right) ^{\prime} = \frac{1}{x} }$$

\begin{aligned} \log_{\textcolor{springgreen}{a}}^{\textcolor{springgreen}{M}} = \frac{\log_{\textcolor{orangered}{b}}^{\textcolor{springgreen}{M}}}{\log_{\textcolor{orangered}{b}}^{\textcolor{springgreen}{a}}} = \frac{\log_{\mathrm{\textcolor{orangered}{e}}}^{\textcolor{springgreen}{M}}}{\log_{\mathrm{\textcolor{orangered}{e}}}^{\textcolor{springgreen}{a}}} = \frac{\ln \textcolor{springgreen}{M}}{\ln \textcolor{springgreen}{a}} \end{aligned}

\begin{aligned} & y \\ \\ = & \ \log_{5} \left( \frac{x}{1-x} \right) \\ \\ = & \ \frac{\ln \frac{x}{1-x}}{\ln 5} \\ \\ = & \ \textcolor{yellow}{ \frac{\ln x – \ln (1-x)}{\ln 5} } \end{aligned}

\begin{aligned} & \frac{\mathrm{d} y}{\mathrm{d} x} \\ \\ = & \left[ \textcolor{yellow}{ \frac{\ln x – \ln (1-x)}{\ln 5} } \right]_{x} ^{\prime} \\ \\ = & \ \frac{1}{\ln 5} \cdot \left[ \ln x – \ln (1-x) \right]_{x} ^{\prime} \\ \\ = & \ \frac{1}{\ln 5} \cdot \left( \frac{1}{x} \textcolor{magenta}{-} \frac{\textcolor{magenta}{-} 1}{1-x} \right) \\ \\ = & \ \frac{1}{\ln 5} \cdot \textcolor{orangered}{ \left( \frac{1}{x} + \frac{1}{1-x} \right) } \\ \\ = & \ \frac{1}{\ln 5} \cdot \textcolor{orangered}{ \frac{1}{x(1-x)} } \\ \\ = & \ \textcolor{springgreen}{\boldsymbol{ \frac{1}{x(1-x) \ln 5 } }} \end{aligned}