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Question-158884




Question Number 158884 by akolade last updated on 10/Nov/21
Answered by ghimisi last updated on 10/Nov/21
log_3 (a+1)=log_4 (a+8)=t⇒  a+1=3^t   a+8=4^t ⇒4^t −3^t =7⇒((4/3))^t =1+7∙((1/3))^t   f(t)=((4/3))^t  ↗ ;g(t)=1+7∙((1/3))^t  ↘  ⇒t=2⇒a=8
$${log}_{\mathrm{3}} \left({a}+\mathrm{1}\right)={log}_{\mathrm{4}} \left({a}+\mathrm{8}\right)={t}\Rightarrow \\ $$$${a}+\mathrm{1}=\mathrm{3}^{{t}} \\ $$$${a}+\mathrm{8}=\mathrm{4}^{{t}} \Rightarrow\mathrm{4}^{{t}} −\mathrm{3}^{{t}} =\mathrm{7}\Rightarrow\left(\frac{\mathrm{4}}{\mathrm{3}}\right)^{{t}} =\mathrm{1}+\mathrm{7}\centerdot\left(\frac{\mathrm{1}}{\mathrm{3}}\right)^{{t}} \\ $$$${f}\left({t}\right)=\left(\frac{\mathrm{4}}{\mathrm{3}}\right)^{{t}} \:\nearrow\:;{g}\left({t}\right)=\mathrm{1}+\mathrm{7}\centerdot\left(\frac{\mathrm{1}}{\mathrm{3}}\right)^{{t}} \:\searrow \\ $$$$\Rightarrow{t}=\mathrm{2}\Rightarrow{a}=\mathrm{8} \\ $$

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