Question

Find the values of ${\log }_5\sqrt{5.\sqrt{5.\sqrt{5.\sqrt{\mathrm{5...}\infty }}}}$

• A. $1$
• B. $-1$
• C. $0$
• D. $2$

Answer 1

The correct option is A $1$

${\log }_5\sqrt{5.\sqrt{5.\sqrt{5.\sqrt{\mathrm{5...}\infty }}}}$

Let $y=\sqrt{5\sqrt{5\sqrt{\mathrm{5...}\infty }}}$

$\Rightarrow y=\sqrt{5y}$

$\Rightarrow y^2=5y$

$\Rightarrow y^2-5y=0$

$\Rightarrow y(y-5)=0$

$\therefore y=0,y=5$

$\Rightarrow y\ne 0\therefore y=5$

$\therefore {\log }_5\sqrt{5.\sqrt{5.\sqrt{5.\sqrt{\mathrm{5...}\infty }}}}$

$={\log }_5\sqrt{5y}$

$={\log }_5\sqrt{25}$ where $y=5$

$={\log }_{5}5$

$=1$ since ${\log }_{a}a=1$