What is the inverse of $y=log_4( x-3) +2x?$ ?

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Answer 1 · 2018-02-26T09:09:35

$x = 1/2(6+W(2^2y-11))$

We can solve this problem using the so called Lambert function $W(cdot)$

$y = lnabs(x-3)/ln4 + 2x rArr y ln4=lnabs(x-3)+2x ln4$

Now making $z = x-3$

$e^(y ln4)=z e^(2(z+3)ln4) = ze^(2z)e^(6 ln4)$ or

$e^((y-6)ln4)=z e^(2z)$ or

$2 e^((y-6)ln4) = 2z e^(2z)$

Now using the equivalence

$Y = X e^X rArr X = W(Y)$

$2z=W(2 e^((y-6)ln4)) rArr z = 1/2 W(2 e^((y-6)ln4))$

and finally

$x = 1/2 W(2 e^((y-6)ln4))+3$ that can be simplified to

$x = 1/2(6+W(2^(2y-11)))$

What is the inverse of y=log_4( x-3) +2x?y=log_4( x-3) +2x? ?