Solve y=ln^3arccossqrt{2^x-1}

$$y = \ln ^ { 3 } \arccos \sqrt { 2 ^ { x - 1 } }$$

Solve for x (complex solution)

$$x=1+\left(2\ln(CosI(e^{\frac{1}{3}y+\frac{1}{3}i\pi }))+4i\pi n_{16}\right)\ln(2)^{-1}\text{, }n_{16}\in \mathrm{Z}$$

$$x=1+\left(2\ln(CosI(e^{\frac{1}{3}y}))+4i\pi n_{2}\right)\ln(2)^{-1}\text{, }n_{2}\in \mathrm{Z}$$

$$x=1+\left(2\ln(CosI(ie^{\frac{1}{3}y+\frac{1}{6}i\pi }))+4i\pi n_{30}\right)\ln(2)^{-1}\text{, }n_{30}\in \mathrm{Z}$$

Solve for y (complex solution)

$$y=\ln(\left(\arccos(\frac{2^{\frac{x+1}{2}}}{2})\right)^{3})$$

$$\nexists n_{1}\in \mathrm{Z}\text{ : }x=\frac{4\pi n_{1}i}{\ln(2)}+1$$

Solve for x

$$x=2\log_{2}\left(\cos(e^{\frac{y}{3}})\right)+1$$

$$\exists n_{2}\in \mathrm{Z}\text{ : }\left(\left(e^{\frac{y}{3}}>2\pi n_{2}+\frac{3\pi }{2}\text{ and }e^{\frac{y}{3}}<2\pi n_{2}+2\pi \right)\text{ or }\left(e^{\frac{y}{3}}>2\pi n_{2}\text{ and }e^{\frac{y}{3}}<2\pi n_{2}+\frac{\pi }{2}\right)\right)\text{ and }\exists n_{1}\in \mathrm{Z}\text{ : }\left(n_{1}>\frac{e^{\frac{y}{3}}-\frac{\pi }{2}}{2\pi }\text{ and }n_{1}<\frac{e^{\frac{y}{3}}+\frac{\pi }{2}}{2\pi }\right)\text{ and }e^{\frac{y}{3}}\leq \pi \text{ and }\exists n_{2}\in \mathrm{Z}\text{ : }\left(\left(n_{2}>\frac{e^{\frac{y}{3}}-\frac{\pi }{2}}{2\pi }\text{ and }n_{2}<\frac{e^{\frac{y}{3}}}{2\pi }\right)\text{ or }\left(n_{2}>\frac{e^{\frac{y}{3}}-2\pi }{2\pi }\text{ and }n_{2}<\frac{e^{\frac{y}{3}}-\frac{3\pi }{2}}{2\pi }\right)\right)\text{ and }\exists n_{1}\in \mathrm{Z}\text{ : }\left(e^{\frac{y}{3}}>2\pi n_{1}-\frac{\pi }{2}\text{ and }e^{\frac{y}{3}}<2\pi n_{1}+\frac{\pi }{2}\right)$$

Solve for y

$$y=3\ln(\arccos(\frac{2^{\frac{x+1}{2}}}{2}))$$

$$x<1$$