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

Feb 26, 2018

$x = \frac{1}{2} \left(6 + W \left({2}^{2} y - 11\right)\right)$

#### Explanation:

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

https://en.wikipedia.org/wiki/Lambert_W_function

$y = \ln \frac{\left\mid x - 3 \right\mid}{\ln} 4 + 2 x \Rightarrow y \ln 4 = \ln \left\mid x - 3 \right\mid + 2 x \ln 4$

Now making $z = x - 3$

${e}^{y \ln 4} = z {e}^{2 \left(z + 3\right) \ln 4} = z {e}^{2 z} {e}^{6 \ln 4}$ or

${e}^{\left(y - 6\right) \ln 4} = z {e}^{2 z}$ or

$2 {e}^{\left(y - 6\right) \ln 4} = 2 z {e}^{2 z}$

Now using the equivalence

$Y = X {e}^{X} \Rightarrow X = W \left(Y\right)$

$2 z = W \left(2 {e}^{\left(y - 6\right) \ln 4}\right) \Rightarrow z = \frac{1}{2} W \left(2 {e}^{\left(y - 6\right) \ln 4}\right)$

and finally

$x = \frac{1}{2} W \left(2 {e}^{\left(y - 6\right) \ln 4}\right) + 3$ that can be simplified to

$x = \frac{1}{2} \left(6 + W \left({2}^{2 y - 11}\right)\right)$