Question

If $a,b\ne 1,$ $ab>0,$ $a\ne b$ and ${\log }_{b}a={\log }_{a}b$, then find $ab.$

Answer 1

Given, ${\log }_{b}a={\log }_{a}b.$

By formula,

$$\begin{gathered} \frac{{\log }_a}{{\log }_b}=\frac{{\log }_b}{{\log }_a} \\ \Rightarrow {(\log a)}^2={(\log b)}^2 \\ \Rightarrow {(\log a)}^2-{(\log b)}^2=0 \\ \Rightarrow (\log a-\log b)(\log a+\log b)=0 \\ \Rightarrow (\log a-\log b)=0or(\log a+\log b)=0 \\ \Rightarrow \log \frac{a}{b}=0or\log ab=0 \\ \Rightarrow \log \frac{a}{b}=\log 1or\log ab=\log 1 \\ \Rightarrow \frac{a}{b}=1orab=1 \\ \Rightarrow a=borab=1 \end{gathered}$$

But given,$a\ne b$

Therefore,$ab=1.$