Question

If $log_e\,5$, $log_e(5^x-1)$ and $log_e$ $\left(5^{x}-\frac{11}{5}\right)$ are in $A.P.$, 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 $\log_5\, 7$

Answer 1

Answer: (A)

$\log_5 \,4$ and $\log_5\, 3$

Answer 2

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