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Question Number 198222 by cortano12 last updated on 14/Oct/23
log _4 (5^x −3^x ) = log _5 (4^x +3^(x ) )
$$\:\:\:\mathrm{log}\:_{\mathrm{4}} \left(\mathrm{5}^{\mathrm{x}} −\mathrm{3}^{\mathrm{x}} \right)\:=\:\mathrm{log}\:_{\mathrm{5}} \left(\mathrm{4}^{\mathrm{x}} +\mathrm{3}^{\mathrm{x}\:} \right) \\ $$
Commented by Rasheed.Sindhi last updated on 14/Oct/23
4^2 +3^2 =5^2 log_5 (4^2 +3^2 )=log_5 5^2 log_5 (4^2 +3^2 )=2......(i) 5^2 −3^2 =4^2 log_4 (5^2 −3^2 )=log_4 4^2 log_4 (5^2 −3^2 )=2......(ii) (i) & (ii): log_5 (4^2 +3^2 )=log_4 (5^2 −3^2 ) Comparing with log_5 (4^x +3^x )=log_4 (5^x −3^x ) yields x=2
$$\mathrm{4}^{\mathrm{2}} +\mathrm{3}^{\mathrm{2}} =\mathrm{5}^{\mathrm{2}} \:\: \\ $$$$\mathrm{log}_{\mathrm{5}} \left(\mathrm{4}^{\mathrm{2}} +\mathrm{3}^{\mathrm{2}} \right)=\mathrm{log}_{\mathrm{5}} \mathrm{5}^{\mathrm{2}} \: \\ $$$$\mathrm{log}_{\mathrm{5}} \left(\mathrm{4}^{\mathrm{2}} +\mathrm{3}^{\mathrm{2}} \right)=\mathrm{2}……\left({i}\right) \\ $$$$ \\ $$$$\mathrm{5}^{\mathrm{2}} −\mathrm{3}^{\mathrm{2}} =\mathrm{4}^{\mathrm{2}} \\ $$$$\mathrm{log}_{\mathrm{4}} \left(\mathrm{5}^{\mathrm{2}} −\mathrm{3}^{\mathrm{2}} \right)=\mathrm{log}_{\mathrm{4}} \mathrm{4}^{\mathrm{2}} \:\: \\ $$$$\mathrm{log}_{\mathrm{4}} \left(\mathrm{5}^{\mathrm{2}} −\mathrm{3}^{\mathrm{2}} \right)=\mathrm{2}……\left({ii}\right) \\ $$$$\left({i}\right)\:\&\:\left({ii}\right): \\ $$$$\:\mathrm{log}_{\mathrm{5}} \left(\mathrm{4}^{\mathrm{2}} +\mathrm{3}^{\mathrm{2}} \right)=\mathrm{log}_{\mathrm{4}} \left(\mathrm{5}^{\mathrm{2}} −\mathrm{3}^{\mathrm{2}} \right) \\ $$$$\mathrm{Comparing}\:\mathrm{with} \\ $$$$\:\mathrm{log}_{\mathrm{5}} \left(\mathrm{4}^{\mathrm{x}} +\mathrm{3}^{\mathrm{x}} \right)=\mathrm{log}_{\mathrm{4}} \left(\mathrm{5}^{\mathrm{x}} −\mathrm{3}^{\mathrm{x}} \right) \\ $$$$\mathrm{yields} \\ $$$$\mathrm{x}=\mathrm{2} \\ $$
Answered by Rasheed.Sindhi last updated on 14/Oct/23
log _4 (5^x −3^x ) = log _5 (4^x +3^(x ) )=k 4^k =5^x −3^x , 5^k =4^x +3^(x ) 4^k +5^k =5^x +4^x ⇒k=x ∴log _4 (5^x −3^x ) = log _5 (4^x +3^(x ) )=x 4^x =5^x −3^x , 5^x =4^x +3^(x ) Both are equivalent to: 4^x +3^(x ) =5^x Which is only true for x=2
$$\:\:\:\mathrm{log}\:_{\mathrm{4}} \left(\mathrm{5}^{\mathrm{x}} −\mathrm{3}^{\mathrm{x}} \right)\:=\:\mathrm{log}\:_{\mathrm{5}} \left(\mathrm{4}^{\mathrm{x}} +\mathrm{3}^{\mathrm{x}\:} \right)=\mathrm{k} \\ $$$$\:\:\:\mathrm{4}^{\mathrm{k}} =\mathrm{5}^{\mathrm{x}} −\mathrm{3}^{\mathrm{x}} \:,\:\:\mathrm{5}^{\mathrm{k}} =\mathrm{4}^{\mathrm{x}} +\mathrm{3}^{\mathrm{x}\:} \\ $$$$\mathrm{4}^{\mathrm{k}} +\mathrm{5}^{\mathrm{k}} =\mathrm{5}^{\mathrm{x}} +\mathrm{4}^{\mathrm{x}} \Rightarrow\mathrm{k}=\mathrm{x} \\ $$$$\:\therefore\mathrm{log}\:_{\mathrm{4}} \left(\mathrm{5}^{\mathrm{x}} −\mathrm{3}^{\mathrm{x}} \right)\:=\:\mathrm{log}\:_{\mathrm{5}} \left(\mathrm{4}^{\mathrm{x}} +\mathrm{3}^{\mathrm{x}\:} \right)=\mathrm{x} \\ $$$$\:\:\:\:\:\:\:\:\mathrm{4}^{\mathrm{x}} =\mathrm{5}^{\mathrm{x}} −\mathrm{3}^{\mathrm{x}} \:,\:\mathrm{5}^{\mathrm{x}} =\mathrm{4}^{\mathrm{x}} +\mathrm{3}^{\mathrm{x}\:} \\ $$$$\mathrm{Both}\:\mathrm{are}\:\mathrm{equivalent}\:\mathrm{to}: \\ $$$$\:\:\:\:\:\:\mathrm{4}^{\mathrm{x}} +\mathrm{3}^{\mathrm{x}\:} =\mathrm{5}^{\mathrm{x}} \\ $$$$\mathrm{Which}\:\mathrm{is}\:\mathrm{only}\:\mathrm{true}\:\mathrm{for} \\ $$$$\:\:\:\:\:\:\:\mathrm{x}=\mathrm{2} \\ $$

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