Question
If $${\log _e}5,{\log _e}\left( {{5^x} - 1} \right){\text{and }}{\log _e}\left( {{5^x} - \frac{{11}}{5}} \right)$$ are in A.P. then the values of $$x$$ are
A.
$${\log _5}4{\text{ and}}\,\,{\log _5}3$$
B.
$${\log _3}4{\text{ and}}\,\,{\log _4}3$$
C.
$${\log _3}4{\text{ and}}\,\,{\log _3}5$$
D.
$${\log _5}6{\text{ and}}\,\,{\log _5}7$$
Answer :
$${\log _5}4{\text{ and}}\,\,{\log _5}3$$
Solution :
$$\eqalign{
& {\log _e}5 + {\log _e}\left( {{5^x} - \frac{{11}}{5}} \right) = 2{\log _e}\left( {{5^x} - 1} \right) \cr
& \Rightarrow {5^{x + 1}} - 11 = {5^{2x}} + 1 - 2 \times {5^x} \cr
& \Rightarrow {5^{2x}} - {7.5^x} + 12 = 0 \cr
& {\text{Let, }}{5^x} = t,{t^2} - 7t + 12 = 0 \cr
& \Rightarrow t = 4,3 \cr
& {5^x} = 4 \cr
& {\log _5}{5^x} = {\log _5}4 \cr
& x = {\log _5}4 \cr
& {5^x} = 3 \cr
& {\log _5}{5^x} = {\log _5}3 \cr
& x = {\log _5}3 \cr} $$