Question-196946

Question Number 196946 by Khalmohmmad last updated on 04/Sep/23

Answered by trisetyo last updated on 04/Sep/23

3^x +2^x =5 3^x =−(2^x −5)∨2^x =−(3^x −5) log_3 (−2^x +5)=x and log_2 (−3^x +5)=x ((log(−2^x +5))/(log(3))) = ((log(−3^x +5))/(log(2))) ((log(−2^x +5))/(log(−3^x +5))) = ((log(3))/(log(2))) ⇒−2^x +5=3 ∨ −3^x +5=2 ⇒2^x =2 ∨ 3^x =3 ⇒x=log_2 (2) ∨ x=log_3 (3) ⇒x=1 ∨ x=1 ∴ x = 1

$$\mathrm{3}^{{x}} +\mathrm{2}^{{x}} =\mathrm{5} \\ $$$$\mathrm{3}^{{x}} =−\left(\mathrm{2}^{{x}} −\mathrm{5}\right)\vee\mathrm{2}^{{x}} =−\left(\mathrm{3}^{{x}} −\mathrm{5}\right) \\ $$$$\mathrm{log}_{\mathrm{3}} \left(−\mathrm{2}^{{x}} +\mathrm{5}\right)={x}\:\:\:\:\mathrm{and}\:\:\:\:\mathrm{log}_{\mathrm{2}} \left(−\mathrm{3}^{{x}} +\mathrm{5}\right)={x} \\ $$$$\frac{\mathrm{log}\left(−\mathrm{2}^{{x}} +\mathrm{5}\right)}{\mathrm{log}\left(\mathrm{3}\right)}\:=\:\frac{\mathrm{log}\left(−\mathrm{3}^{{x}} +\mathrm{5}\right)}{\mathrm{log}\left(\mathrm{2}\right)} \\ $$$$\frac{\mathrm{log}\left(−\mathrm{2}^{{x}} +\mathrm{5}\right)}{\mathrm{log}\left(−\mathrm{3}^{{x}} +\mathrm{5}\right)}\:=\:\frac{\mathrm{log}\left(\mathrm{3}\right)}{\mathrm{log}\left(\mathrm{2}\right)} \\ $$$$\Rightarrow−\mathrm{2}^{{x}} +\mathrm{5}=\mathrm{3}\:\:\vee\:\:−\mathrm{3}^{{x}} +\mathrm{5}=\mathrm{2} \\ $$$$\Rightarrow\mathrm{2}^{{x}} =\mathrm{2}\:\:\vee\:\:\mathrm{3}^{{x}} =\mathrm{3} \\ $$$$\Rightarrow{x}=\mathrm{log}_{\mathrm{2}} \left(\mathrm{2}\right)\:\:\vee\:\:{x}=\mathrm{log}_{\mathrm{3}} \left(\mathrm{3}\right) \\ $$$$\Rightarrow{x}=\mathrm{1}\:\:\vee\:\:{x}=\mathrm{1} \\ $$$$\therefore\:\:{x}\:=\:\mathrm{1} \\ $$

Commented by Frix last updated on 04/Sep/23

Nonsense. At least test your solution 3^x −2^x =3^1 −2^1 =1≠5

$$\mathrm{Nonsense}. \\ $$$$\mathrm{At}\:\mathrm{least}\:\mathrm{test}\:\mathrm{your}\:\mathrm{solution} \\ $$$$\mathrm{3}^{{x}} −\mathrm{2}^{{x}} =\mathrm{3}^{\mathrm{1}} −\mathrm{2}^{\mathrm{1}} =\mathrm{1}\neq\mathrm{5} \\ $$

Commented by trisetyo last updated on 04/Sep/23

oh i′m sorry... its 3^x −2^x =5 my answer 3^x +2^x =5 it was my mistake, sorry..

$$ \\ $$$$\mathrm{oh}\:\mathrm{i}'\mathrm{m}\:\mathrm{sorry}…\:\mathrm{its}\:\mathrm{3}^{{x}} −\mathrm{2}^{{x}} =\mathrm{5} \\ $$$$\mathrm{my}\:\mathrm{answer}\:\mathrm{3}^{{x}} +\mathrm{2}^{{x}} =\mathrm{5} \\ $$$$\mathrm{it}\:\mathrm{was}\:\mathrm{my}\:\mathrm{mistake},\:{sorry}.. \\ $$

Answered by Frix last updated on 04/Sep/23

$$\mathrm{3}^{\mathrm{2}} −\mathrm{2}^{\mathrm{2}} =\mathrm{9}−\mathrm{4}=\mathrm{5} \\ $$$${x}=\mathrm{2} \\ $$

Commented by trisetyo last updated on 04/Sep/23

$$ \\ $$$${verry}\:{simple}\:{solution}\:{XD} \\ $$

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