Evaluate the following determinants:
bar of  (log _ab   1)
             (1        log_ba)
           

 

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{By} \mathrm{Base} \mathrm{change} \mathrm{formula} \mathrm{we} \mathrm{have} \\ {\log }_{\mathrm{p}}\mathrm{q}=\frac{{\log }_{10}\mathrm{q}}{{\log }_{10}\mathrm{p}}...\left(\mathrm{i}\right) \\ \mathrm{Hence} \mathrm{we} \mathrm{have} \\ △=\left|\begin{array}{cc}{\log }_{\mathrm{a}}\mathrm{b}& 1\\ 1& {\log }_{\mathrm{b}}\mathrm{a}\end{array}\right| \\ ={\log }_{\mathrm{a}}\mathrm{b}\times {\log }_{\mathrm{b}}\mathrm{a}-1 \\ \mathrm{Use} \left(\mathrm{i}\right) \\ =\frac{{\log }_{10}\mathrm{b}}{{\log }_{10}\mathrm{a}}\times \frac{{\log }_{10}\mathrm{a}}{{\log }_{10}\mathrm{b}}-1 \\ =1-1 \\ =0 \left[\mathrm{Answer}\right] \end{gathered}$$

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