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8. Sequences and Series
easy
જો $x,\;y,\;z$ એ સમગુણોતર શ્નેણીમાંં હોય અને ${a^x} = {b^y} = {c^z}$ તે
A
${\log _a}c = {\log _b}a$
B
${\log _b}a = {\log _c}b$
C
${\log _c}b = {\log _a}c$
D
એકપણ નહિ
(IIT-1966) (IIT-1968)
Solution
(b) $x,\;y,\;z$ are in $G.P.$, then ${y^2} = x\;.\;z$
Now ${a^x} = {b^y} = {c^z} = m$
$ \Rightarrow $ $x{\log _e}a = y{\log _e}b = z{\log _e}c = {\log _e}m$
$ \Rightarrow $ $x = {\log _a}m,\;y = {\log _b}m,z = {\log _c}m$
Again as $x,\;y,\;z$ are in $G.P.$,
so $\frac{{y}}{x}=\frac{{z}}{y}$
$ \Rightarrow $$\frac{{{{\log }_b}m}}{{{{\log }_a}m}} = \frac{{{{\log }_c}m}}{{{{\log }_b}m}}$
$ \Rightarrow $${\log _b}a = {\log _c}b$.
Standard 11
Mathematics
