Question

If ${\log }_{e}5,{\log }_e(5^x-1)$ and ${\log }_e(5^x-\frac{11}{5})$ are in $AP,$ then the values of $x$ are :

• A. ${\log }_{5}4$ and ${\log }_{5}3$
• B. ${\log }_{3}4$ and ${\log }_{4}3$
• C. ${\log }_{3}4$ and ${\log }_{3}5$
• D. ${\log }_{5}6$ and ${\log }_{5}7$
• E. $12,6$

Answer 1

The correct option is A ${\log }_{5}4$ and ${\log }_{5}3$

Since, ${\log }_{e}5,{\log }_e(5^x-1)$ and ${\log }_e(5^x-\frac{11}{5})$ are in $AP.$
Therefore, $2{\log }_e(5^x-1)={\log }_{e}5+{\log }_e(5^x-\frac{11}{5})$
$\Rightarrow (5^x-1{)}^2=5(5^x-\frac{11}{5})$
$\Rightarrow 5^{2x}+1-2\times 5^x=5\times 5^x-11$
$\Rightarrow 5^{2x}-7\times 5^x+12=0$
$\Rightarrow 5^{2x}-4\times 5^x-3\times 5^x+12=0$
$\Rightarrow (5^x-4)(5^x-3)=0$
$\Rightarrow 5^x=4,5^x=3$
$\Rightarrow x={\log }_{5}4,x={\log }_{5}3$