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Question Number 5902 by Ashis last updated on 04/Jun/16

solve x 6^(2x+4) =3^(3x) 2^(x+8)

$${solve}\:{x} \\ $$$$\mathrm{6}^{\mathrm{2}{x}+\mathrm{4}} \:\:=\mathrm{3}^{\mathrm{3}{x}} \:\mathrm{2}^{{x}+\mathrm{8}} \\ $$

Commented by prakash jain last updated on 04/Jun/16

6^(2x) 6^4 =3^(3x) 2^x 2^8 (2∙3)^(2x) 3^4 2^4 =3^(3x) 2^x 2^8 2^(2x) 3^(2x) 3^4 2^4 =3^(3x) 2^x 2^8 ((2^(2x) 3^(2x) )/(3^(3x) 2^x ))=(2^8 /(3^4 2^4 )) (2^x /3^x )=(2^4 /3^4 ) ((2/3))^x =((2/3))^4 x=4

$$\mathrm{6}^{\mathrm{2}{x}} \mathrm{6}^{\mathrm{4}} =\mathrm{3}^{\mathrm{3}{x}} \mathrm{2}^{{x}} \mathrm{2}^{\mathrm{8}} \\ $$$$\left(\mathrm{2}\centerdot\mathrm{3}\right)^{\mathrm{2}{x}} \mathrm{3}^{\mathrm{4}} \mathrm{2}^{\mathrm{4}} =\mathrm{3}^{\mathrm{3}{x}} \mathrm{2}^{{x}} \mathrm{2}^{\mathrm{8}} \\ $$$$\mathrm{2}^{\mathrm{2}{x}} \mathrm{3}^{\mathrm{2}{x}} \mathrm{3}^{\mathrm{4}} \mathrm{2}^{\mathrm{4}} =\mathrm{3}^{\mathrm{3}{x}} \mathrm{2}^{{x}} \mathrm{2}^{\mathrm{8}} \\ $$$$\frac{\mathrm{2}^{\mathrm{2}{x}} \mathrm{3}^{\mathrm{2}{x}} }{\mathrm{3}^{\mathrm{3}{x}} \mathrm{2}^{{x}} }=\frac{\mathrm{2}^{\mathrm{8}} }{\mathrm{3}^{\mathrm{4}} \mathrm{2}^{\mathrm{4}} } \\ $$$$\frac{\mathrm{2}^{{x}} }{\mathrm{3}^{{x}} }=\frac{\mathrm{2}^{\mathrm{4}} }{\mathrm{3}^{\mathrm{4}} } \\ $$$$\left(\frac{\mathrm{2}}{\mathrm{3}}\right)^{{x}} =\left(\frac{\mathrm{2}}{\mathrm{3}}\right)^{\mathrm{4}} \\ $$$${x}=\mathrm{4} \\ $$

Answered by sanusihammed last updated on 04/Jun/16

Solution 6^(2x + 4) = 3^(3x) 2^(x+8) Take the logarithm of both sides log6^(2x+4) = log(3^(3x) 2^(x+8) ) (2x+4)log6 = log3^(3x) + log2^(x+8) (2x+4)log6 = 3xlog3 + (x+8)log2 (2x+4)0.7782 = 3x(0.4771) + (x+8)(0.3010) 1.5564x + 3.1128 = 1.4313x + 0.3010x + 2.408 1.5564x + 3.1128 = 1.7323x + 2.408 collect the like terms 1.7323x−1.5564x = 3.1128−2.408 0.1759x = 0.7048 x = ((0.7048)/(0.1759)) x = 4.0068 Approximately x = 4

$${Solution} \\ $$$$ \\ $$$$\mathrm{6}^{\mathrm{2}{x}\:+\:\mathrm{4}} \:=\:\mathrm{3}^{\mathrm{3}{x}} \mathrm{2}^{{x}+\mathrm{8}} \\ $$$${Take}\:{the}\:{logarithm}\:{of}\:{both}\:{sides} \\ $$$${log}\mathrm{6}^{\mathrm{2}{x}+\mathrm{4}} \:=\:{log}\left(\mathrm{3}^{\mathrm{3}{x}} \mathrm{2}^{{x}+\mathrm{8}} \right) \\ $$$$\left(\mathrm{2}{x}+\mathrm{4}\right){log}\mathrm{6}\:=\:{log}\mathrm{3}^{\mathrm{3}{x}} \:+\:{log}\mathrm{2}^{{x}+\mathrm{8}} \\ $$$$\left(\mathrm{2}{x}+\mathrm{4}\right){log}\mathrm{6}\:=\:\mathrm{3}{xlog}\mathrm{3}\:+\:\left({x}+\mathrm{8}\right){log}\mathrm{2} \\ $$$$\left(\mathrm{2}{x}+\mathrm{4}\right)\mathrm{0}.\mathrm{7782}\:=\:\mathrm{3}{x}\left(\mathrm{0}.\mathrm{4771}\right)\:+\:\left({x}+\mathrm{8}\right)\left(\mathrm{0}.\mathrm{3010}\right) \\ $$$$\mathrm{1}.\mathrm{5564}{x}\:+\:\mathrm{3}.\mathrm{1128}\:=\:\mathrm{1}.\mathrm{4313}{x}\:+\:\mathrm{0}.\mathrm{3010}{x}\:+\:\mathrm{2}.\mathrm{408} \\ $$$$\mathrm{1}.\mathrm{5564}{x}\:+\:\mathrm{3}.\mathrm{1128}\:=\:\mathrm{1}.\mathrm{7323}{x}\:+\:\mathrm{2}.\mathrm{408} \\ $$$${collect}\:{the}\:{like}\:{terms} \\ $$$$\mathrm{1}.\mathrm{7323}{x}−\mathrm{1}.\mathrm{5564}{x}\:=\:\mathrm{3}.\mathrm{1128}−\mathrm{2}.\mathrm{408} \\ $$$$\mathrm{0}.\mathrm{1759}{x}\:=\:\mathrm{0}.\mathrm{7048} \\ $$$${x}\:=\:\frac{\mathrm{0}.\mathrm{7048}}{\mathrm{0}.\mathrm{1759}} \\ $$$${x}\:=\:\mathrm{4}.\mathrm{0068} \\ $$$${Approximately} \\ $$$${x}\:=\:\mathrm{4} \\ $$$$ \\ $$$$ \\ $$

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