Question

What is the formula of $\log \left(\frac{1}{x}\right)$?

Answer 1

Determine the formula of $\log \left(\frac{1}{x}\right)$.

$$\begin{gathered} \begin{array}{rcl}\log \left(\frac{1}{x}\right)& =& \log \left(x^{-1}\right)\\ & \Rightarrow & \log \left(\frac{1}{x}\right)=\log {\left(x\right)}^{-1}\end{array} \\ \begin{array}{rcl}& \Rightarrow & \log \left(\frac{1}{x}\right)=-1.\log \left(x\right)\end{array}\left[\mathbf{\because }\mathit{l}\mathit{o}\mathit{g}\mathbf{}{\left(x\right)}^{\mathbf{a}}\mathbf{=}\mathit{a}\mathit{l}\mathit{o}\mathit{g}\mathbf{}\left(x\right)\right] \\ \begin{array}{rcl}& \Rightarrow & \log \left(\frac{1}{x}\right)=-\log \left(x\right)\end{array} \end{gathered}$$

Hence, the formula is $\begin{array}{rcl}\log \left(\frac{1}{x}\right)& =& -\log \left(x\right)\end{array}$.