Question

If $a,b\ne 1,ab>0,a\ne band{\log }_{b}a={\log }_{a}b$ then ab = ?

• A. 1/2
• B. 1
• C. 2
• D. 10

Answer 1

The correct option is B 1

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

$\Rightarrow \frac{\log a}{\log b}=\frac{\log b}{\log a}$

$\Rightarrow (\log a{)}^2=(\log b{)}^2$

$\Rightarrow (\log a{)}^2-(\log b{)}^2=0$

$\Rightarrow (\log a-\log b)(\log a+\log b)=0$

$\Rightarrow \log a-\log b=0$ OR $\log a+\log b=0$

$\Rightarrow \log \frac{a}{b}=0$ OR $\log ab=0$

$\Rightarrow \log \frac{a}{b}=\log 1$ OR $\log ab=\log 1$

$\Rightarrow \frac{a}{b}=1$ OR $ab=1$

$\Rightarrow a=b$ OR $ab=1$

but given $a\ne b$

$\therefore ab=1$

Option B is correct.