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Question Number 99050 by bemath last updated on 18/Jun/20

5^(5−3x) + 2^(x+5) = 5^(7−3x) −2^(x+6)

$$\mathrm{5}^{\mathrm{5}−\mathrm{3}{x}} \:+\:\mathrm{2}^{{x}+\mathrm{5}} \:=\:\mathrm{5}^{\mathrm{7}−\mathrm{3}{x}} \:−\mathrm{2}^{{x}+\mathrm{6}} \: \\ $$

Commented by bobhans last updated on 18/Jun/20

(5^5 /5^(3x) ) + 32.2^x = (5^7 /5^(3x) )−64.2^x 96.2^x = ((5^5 (5^2 −1))/5^(3x) ) = ((24.5^5 )/5^(3x) ) 4.2^x = 5^(5−3x) ⇒2^(2+x) = 5^(5−3x) (2+x)ln(2) = (5−3x) ln(5) x ln(2)+3x ln(5)= 5ln(5)−2ln(2) x(0.693+4.828)= 8.047−1.386 x = ((6.661)/(5.521)) ≈ 1.206

$$\frac{\mathrm{5}^{\mathrm{5}} }{\mathrm{5}^{\mathrm{3x}} }\:+\:\mathrm{32}.\mathrm{2}^{\mathrm{x}} \:=\:\frac{\mathrm{5}^{\mathrm{7}} }{\mathrm{5}^{\mathrm{3x}} }−\mathrm{64}.\mathrm{2}^{\mathrm{x}} \\ $$$$\mathrm{96}.\mathrm{2}^{\mathrm{x}} \:=\:\frac{\mathrm{5}^{\mathrm{5}} \left(\mathrm{5}^{\mathrm{2}} −\mathrm{1}\right)}{\mathrm{5}^{\mathrm{3x}} }\:=\:\frac{\mathrm{24}.\mathrm{5}^{\mathrm{5}} }{\mathrm{5}^{\mathrm{3x}} } \\ $$$$\mathrm{4}.\mathrm{2}^{\mathrm{x}} \:=\:\mathrm{5}^{\mathrm{5}−\mathrm{3x}} \:\Rightarrow\mathrm{2}^{\mathrm{2}+\mathrm{x}} =\:\mathrm{5}^{\mathrm{5}−\mathrm{3x}} \\ $$$$\left(\mathrm{2}+\mathrm{x}\right)\mathrm{ln}\left(\mathrm{2}\right)\:=\:\left(\mathrm{5}−\mathrm{3x}\right)\:\mathrm{ln}\left(\mathrm{5}\right) \\ $$$$\mathrm{x}\:\mathrm{ln}\left(\mathrm{2}\right)+\mathrm{3x}\:\mathrm{ln}\left(\mathrm{5}\right)=\:\mathrm{5ln}\left(\mathrm{5}\right)−\mathrm{2ln}\left(\mathrm{2}\right) \\ $$$$\mathrm{x}\left(\mathrm{0}.\mathrm{693}+\mathrm{4}.\mathrm{828}\right)=\:\mathrm{8}.\mathrm{047}−\mathrm{1}.\mathrm{386} \\ $$$$\mathrm{x}\:=\:\frac{\mathrm{6}.\mathrm{661}}{\mathrm{5}.\mathrm{521}}\:\approx\:\mathrm{1}.\mathrm{206} \\ $$

Commented by bemath last updated on 18/Jun/20

thank you both sir

$$\mathrm{thank}\:\mathrm{you}\:\mathrm{both}\:\mathrm{sir} \\ $$$$ \\ $$

Answered by Rasheed.Sindhi last updated on 18/Jun/20

5^(5−3x) + 2^(x+5) = 5^(7−3x) −2^(x+6) 2^(x+5) + 2^(x+6) = 5^(7−3x) −5^(5−3x) 2^(x+5) (1+2)=5^(5−3x) (5^2 −1) ((2^(x+5) ×5^(3x) )/5^5 )=((24)/3) 2^5 .2^x .125^x =((24.5^5 )/3) (250)^x =((24.5^5 )/(3.2^5 )) (250)^x =((8.5^5 )/2^5 )=(5^5 /2^2 ) x=((5log5−2log2)/(log250))≈1.2064

$$\mathrm{5}^{\mathrm{5}−\mathrm{3}{x}} \:+\:\mathrm{2}^{{x}+\mathrm{5}} \:=\:\mathrm{5}^{\mathrm{7}−\mathrm{3}{x}} \:−\mathrm{2}^{{x}+\mathrm{6}} \: \\ $$$$\:\:\:\:\:\:\:\mathrm{2}^{{x}+\mathrm{5}} +\:\mathrm{2}^{{x}+\mathrm{6}} \:=\:\mathrm{5}^{\mathrm{7}−\mathrm{3}{x}} \:−\mathrm{5}^{\mathrm{5}−\mathrm{3}{x}} \: \\ $$$$\:\:\:\:\:\mathrm{2}^{{x}+\mathrm{5}} \left(\mathrm{1}+\mathrm{2}\right)=\mathrm{5}^{\mathrm{5}−\mathrm{3}{x}} \left(\mathrm{5}^{\mathrm{2}} −\mathrm{1}\right)\:\: \\ $$$$\:\:\:\:\frac{\mathrm{2}^{{x}+\mathrm{5}} ×\mathrm{5}^{\mathrm{3}{x}} }{\mathrm{5}^{\mathrm{5}} }=\frac{\mathrm{24}}{\mathrm{3}} \\ $$$$\:\:\:\:\mathrm{2}^{\mathrm{5}} .\mathrm{2}^{{x}} .\mathrm{125}^{{x}} =\frac{\mathrm{24}.\mathrm{5}^{\mathrm{5}} }{\mathrm{3}} \\ $$$$\:\:\:\:\:\:\left(\mathrm{250}\right)^{{x}} =\frac{\mathrm{24}.\mathrm{5}^{\mathrm{5}} }{\mathrm{3}.\mathrm{2}^{\mathrm{5}} } \\ $$$$\:\:\:\:\:\:\left(\mathrm{250}\right)^{{x}} =\frac{\mathrm{8}.\mathrm{5}^{\mathrm{5}} }{\mathrm{2}^{\mathrm{5}} }=\frac{\mathrm{5}^{\mathrm{5}} }{\mathrm{2}^{\mathrm{2}} } \\ $$$$\:\:\:\:\:\:{x}=\frac{\mathrm{5log5}−\mathrm{2log2}}{\mathrm{log250}}\approx\mathrm{1}.\mathrm{2064} \\ $$

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