Solve lg(4^-1*2^sqrt{x}-1)-1=lg(2^sqrt{x-2}+2)-2lg2

Solve for x (complex solution)

x=\left(4\ln(2)+i\pi +2i\pi n_{27}\right)^{2}\ln(2)^{-2}

n_{27}\in \mathrm{Z}

\left(\left(\left(\left(1+\left(-1\right)sign(\pi +2\pi n_{27})\right)\pi +ArcTanI(\frac{1}{4}\ln(2)^{-1}\left(\pi +2\pi n_{27}\right))<\pi \text{ and }Im(\ln(\frac{1}{4}\times 2^{4+\left(i\pi +2i\pi n_{27}\right)\ln(2)^{-1}}-1))+\left(-1\right)Im(\ln(2+\frac{1}{4}\times 2^{4+\left(i\pi +2i\pi n_{27}\right)\ln(2)^{-1}}))=0\right)\text{ and }\left(\nexists n_{1}\in \mathrm{Z}\text{ : }\left(n_{1}=\frac{1}{2}i\left(2\ln(2)+\left(16in_{27}\ln(2)\pi +\left(-4\right)n_{27}\pi ^{2}+\left(-4\right)n_{27}^{2}\pi ^{2}+16\ln(2)^{2}+8i\pi \ln(2)+\left(-1\right)\pi ^{2}\right)^{\frac{1}{2}}\right)\pi ^{-1}\text{ or }n_{1}=\frac{1}{2}i\left(2\ln(2)+\left(-1\right)\left(16in_{27}\ln(2)\pi +\left(-4\right)n_{27}\pi ^{2}+\left(-4\right)n_{27}^{2}\pi ^{2}+16\ln(2)^{2}+8i\pi \ln(2)+\left(-1\right)\pi ^{2}\right)^{\frac{1}{2}}\right)\pi ^{-1}\right)\text{ and }\nexists n_{2}\in \mathrm{Z}\text{ : }\left(n_{2}=\frac{1}{2}i\left(i\pi +3\ln(2)+\left(16i\ln(2)\pi n_{27}+\left(-4\right)\pi ^{2}n_{27}+\left(-4\right)\pi ^{2}n_{27}^{2}+16\ln(2)^{2}+8i\pi \ln(2)+\left(-1\right)\pi ^{2}\right)^{\frac{1}{2}}\right)\pi ^{-1}\text{ or }n_{2}=\frac{1}{2}i\left(i\pi +3\ln(2)+\left(-1\right)\left(16i\ln(2)\pi n_{27}+\left(-4\right)\pi ^{2}n_{27}+\left(-4\right)\pi ^{2}n_{27}^{2}+16\ln(2)^{2}+8i\pi \ln(2)+\left(-1\right)\pi ^{2}\right)^{\frac{1}{2}}\right)\pi ^{-1}\right)\right)\right)\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }\left(4\ln(2)+i\pi +2i\pi n_{27}\right)^{2}\ln(2)^{-2}=4\left(\ln(2)+i\pi n_{1}\right)^{2}\ln(2)^{-2}\right)\text{ and }\nexists n_{2}\in \mathrm{Z}\text{ : }\left(4\ln(2)+i\pi +2i\pi n_{27}\right)^{2}\ln(2)^{-2}=\left(3\ln(2)+i\pi +2i\pi n_{2}\right)^{2}\ln(2)^{-2}