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જો $x, y, z$ સમગુણોત્તર શ્રેણીમાં અને $a^x = b^y = c^z$ હોય, તો . . . . . .
$log_b\ a = log_a\ c$
$log_c\ b = log_a\ c$
$log_b\ a = log_c\ b$
આપેલ પૈકી એકપણ નહિ.
Solution
$x, y, z$ સમગુણોત્તર શ્રેણીમાં છે. $⇒ y^2 = xz$
ધારો, ${{\text{a}}^{\text{x}}}={{\text{b}}^{\text{y}}}={{\text{c}}^{\text{z}}}=\text{ }\!\!\lambda\!\!\text{ (say)}\,$
$\Rightarrow \,\,\text{xloga}=\text{ylogb}=\text{zlogc}=\text{log }\!\!\lambda\!\!\text{ }\,$
$\Rightarrow \,\,\text{x}=\frac{\text{log }\!\!\lambda\!\!\text{ }}{\text{loga}}\text{,}\,\text{y}=\frac{\text{log}\,\text{ }\!\!\lambda\!\!\text{ }}{\text{logb}}\text{,}\,\,\text{z}=\frac{\text{log}\,\text{ }\!\!\lambda\!\!\text{ }}{\text{logc}}$
સમીકરણ (i) માં $x, y, z$ મૂકતાં
${{\left( \frac{\text{log }\!\!\lambda\!\!\text{ }}{\text{logb}} \right)}^{\text{2}}}=\frac{\text{log }\!\!\lambda\!\!\text{ }}{\text{loga}}\text{.}\frac{\text{log }\!\!\lambda\!\!\text{ }}{\text{logc}}$
${{\text{(logb)}}^{\text{2}}}=\text{loga}\,\text{.}\,\text{logc}$ અથવા $ \text{lo}{{\text{g}}_{\text{a}}}\text{b}=\text{lo}{{\text{g}}_{\text{b}}}\text{c}$
$\Rightarrow \,\text{lo}{{\text{g}}_{\text{b}}}\text{a}=\text{lo}{{\text{g}}_{\text{c}}}\text{b}$
