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8-x-1-3-4-x-1-solve-for-x-Any-idea-

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Question Number 22260 by Bruce Lee last updated on 14/Oct/17
8^(x−1) +(3/4)x=1 solve for x Any idea?
8x−1+34x=1solveforxAnyidea?
Answered by mrW1 last updated on 15/Oct/17
8^(x−1) =((4−3x)/4) 2^(3x−3) =−((3x−4)/4) 2×2^(3x−4) =−((3x−4)/4) 2^(3x−4) =−((3x−4)/8) let t=3x−4 ⇒2^t =−(t/8) ⇒e^(tln 2) =−(t/8) ⇒((−te^(−tln 2) )/8)=1 ⇒((−tln 2e^(−tln 2) )/(8ln 2))=1 ⇒(−tln 2)e^((−tln 2)) =8ln 2 ⇒−tln 2=W(8ln 2) ⇒t=−((W(8ln 2))/(ln 2))=3x−4 ⇒x=(1/3)[4−((W(8ln 2))/(ln 2))]=(1/3)[4−2]=(2/3) this method works for any other values: a^(x−1) +bx=c ⇒a^(x−1) =−b(x−(c/b)) ⇒a^(x−(c/b)+(c/b)−1) =−b(x−(c/b)) ⇒a^((c/b)−1) ×a^(x−(c/b)) =−b(x−(c/b)) ⇒a^((c/b)−1) ×e^((x−(c/b))ln a) =−b(x−(c/b)) ⇒a^((c/b)−1) =−b(x−(c/b))e^(−(x−(c/b))ln a) ⇒((a^((c/b)−1) ln a)/b)=[−(x−(c/b))ln a]e^(−(x−(c/b))ln a) ⇒−(x−(c/b))ln a=W(((a^((c/b)−1) ln a)/b)) ⇒x=(c/b)−((W(((a^((c/b)−1) ln a)/b)))/(ln a))
8x−1=4−3x423x−3=−3x−442×23x−4=−3x−4423x−4=−3x−48lett=3x−4⇒2t=−t8⇒etln2=−t8⇒−te−tln28=1⇒−tln2e−tln28ln2=1⇒(−tln2)e(−tln2)=8ln2⇒−tln2=W(8ln2)⇒t=−W(8ln2)ln2=3x−4⇒x=13[4−W(8ln2)ln2]=13[4−2]=23thismethodworksforanyothervalues:ax−1+bx=c⇒ax−1=−b(x−cb)⇒ax−cb+cb−1=−b(x−cb)⇒acb−1×ax−cb=−b(x−cb)⇒acb−1×e(x−cb)lna=−b(x−cb)⇒acb−1=−b(x−cb)e−(x−cb)lna⇒acb−1lnab=[−(x−cb)lna]e−(x−cb)lna⇒−(x−cb)lna=W(acb−1lnab)⇒x=cb−W(acb−1lnab)lna
Answered by mrW1 last updated on 15/Oct/17
8^(x−1) +(3/4)x=1 ⇒(8^x /8)+((3x)/4)= ⇒(2^(3x) /8)+((3x)/4)=1 try with 3x=2 (2^2 /8)+(2/4)=(1/2)+(1/2)= 1 ⇒ok ⇒3x=2 is correct ⇒x=(2/3) this method works only in very special cases.
8x−1+34x=1⇒8x8+3x4=⇒23x8+3x4=1trywith3x=2228+24=12+12=1⇒ok⇒3x=2iscorrect⇒x=23thismethodworksonlyinveryspecialcases.

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