Leysa {l}{2^x=3}{4^y=5}{z=2^x-2y}{a=z}{b=a}{c=b}{text{Solvefor}dtext{where}}{d=c}

$$\left. \begin{array} { l } { 2 ^ {x} = 3 }\\ { 4 ^ {y} = 5 }\\ { z = 2 ^ {x - 2 y} }\\ { a = z }\\ { b = a }\\ { c = b }\\ { \text{Solve for } d \text{ where} } \\ { d = c } \end{array} \right.$$

Leystu fyrir x, y, z, a, b, c, d (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(2)}+\log_{2}\left(3\right)$$

$$n_{1}\in \mathrm{Z}$$

$$y=\frac{\pi n_{2}i}{\ln(2)}+\frac{\log_{2}\left(5\right)}{2}$$

$$n_{2}\in \mathrm{Z}$$

$$z\in \cup n_{1},2^{\frac{2\pi n_{1}i}{\ln(2)}-\frac{2\pi n_{2}i}{\ln(2)}+\log_{2}\left(\frac{3}{5}\right)}$$

$$n_{2}\in \mathrm{Z}$$

$$a\in \cup n_{2},\cup n_{1},\cup n_{2},\cup n_{1},2^{\frac{2\pi n_{1}i}{\ln(2)}-\frac{2\pi n_{2}i}{\ln(2)}+\log_{2}\left(\frac{3}{5}\right)}$$

$$z=2^{\frac{2\pi n_{1}i}{\ln(2)}-\frac{2\pi n_{2}i}{\ln(2)}+\frac{\ln(\frac{3}{5})}{\ln(2)}}$$

$$n_{2}\in \mathrm{Z}$$

$$b\in \cup n_{2},\cup n_{2},\cup n_{1},\cup n_{2},\cup n_{2},\cup n_{1},\cup n_{2},\cup n_{1},2^{\frac{2\pi n_{1}i}{\ln(2)}-\frac{2\pi n_{2}i}{\ln(2)}+\log_{2}\left(\frac{3}{5}\right)}$$

$$a=2^{\frac{2\pi n_{1}i}{\ln(2)}-\frac{2\pi n_{2}i}{\ln(2)}+\frac{\ln(\frac{3}{5})}{\ln(2)}}\text{ and }z=2^{\frac{2\pi n_{1}i}{\ln(2)}-\frac{2\pi n_{2}i}{\ln(2)}+\frac{\ln(\frac{3}{5})}{\ln(2)}}$$

$$\exists n_{2}\in \mathrm{Z}\text{ : }\left(\exists n_{1}\in \mathrm{Z}\text{ : }z=2^{\frac{2\pi n_{1}i}{\ln(2)}-\frac{2\pi n_{2}i}{\ln(2)}+\frac{\ln(\frac{3}{5})}{\ln(2)}}\right)$$

$$n_{2}\in \mathrm{Z}$$

$$c\in \cup n_{2},\cup n_{2},\cup n_{2},\cup n_{1},\cup n_{2},\cup n_{2},\cup n_{2},\cup n_{1},\cup n_{2},\cup n_{2},\cup n_{1},\cup n_{2},\cup n_{1},2^{\frac{2\pi n_{1}i}{\ln(2)}-\frac{2\pi n_{2}i}{\ln(2)}+\log_{2}\left(\frac{3}{5}\right)}$$

$$b=2^{\frac{2\pi n_{1}i}{\ln(2)}-\frac{2\pi n_{2}i}{\ln(2)}+\frac{\ln(\frac{3}{5})}{\ln(2)}}\text{ and }a=2^{\frac{2\pi n_{1}i}{\ln(2)}-\frac{2\pi n_{2}i}{\ln(2)}+\frac{\ln(\frac{3}{5})}{\ln(2)}}\text{ and }z=2^{\frac{2\pi n_{1}i}{\ln(2)}-\frac{2\pi n_{2}i}{\ln(2)}+\frac{\ln(\frac{3}{5})}{\ln(2)}}$$

$$\exists n_{2}\in \mathrm{Z}\text{ : }\left(\exists n_{1}\in \mathrm{Z}\text{ : }z=2^{\frac{2\pi n_{1}i}{\ln(2)}-\frac{2\pi n_{2}i}{\ln(2)}+\frac{\ln(\frac{3}{5})}{\ln(2)}}\right)$$

$$\exists n_{2}\in \mathrm{Z}\text{ : }\left(\exists n_{1}\in \mathrm{Z}\text{ : }z=2^{\frac{2\pi n_{1}i}{\ln(2)}-\frac{2\pi n_{2}i}{\ln(2)}+\frac{\ln(\frac{3}{5})}{\ln(2)}}\right)\text{ and }\exists n_{2}\in \mathrm{Z}\text{ : }\left(\exists n_{2}\in \mathrm{Z}\text{ : }\left(\exists n_{1}\in \mathrm{Z}\text{ : }\left(\exists n_{2}\in \mathrm{Z}\text{ : }\left(\exists n_{1}\in \mathrm{Z}\text{ : }a=2^{\frac{2\pi n_{1}i}{\ln(2)}-\frac{2\pi n_{2}i}{\ln(2)}+\frac{\ln(\frac{3}{5})}{\ln(2)}}\right)\right)\right)\right)$$

$$n_{2}\in \mathrm{Z}$$

$$d\in \cup n_{2},\cup n_{2},\cup n_{2},\cup n_{2},\cup n_{1},\cup n_{2},\cup n_{2},\cup n_{2},\cup n_{2},\cup n_{1},\cup n_{2},\cup n_{2},\cup n_{2},\cup n_{1},\cup n_{2},\cup n_{2},\cup n_{1},\cup n_{2},\cup n_{1},2^{\frac{2\pi n_{1}i}{\ln(2)}-\frac{2\pi n_{2}i}{\ln(2)}+\log_{2}\left(\frac{3}{5}\right)}$$

$$c=2^{\frac{2\pi n_{1}i}{\ln(2)}-\frac{2\pi n_{2}i}{\ln(2)}+\frac{\ln(\frac{3}{5})}{\ln(2)}}\text{ and }b=2^{\frac{2\pi n_{1}i}{\ln(2)}-\frac{2\pi n_{2}i}{\ln(2)}+\frac{\ln(\frac{3}{5})}{\ln(2)}}\text{ and }a=2^{\frac{2\pi n_{1}i}{\ln(2)}-\frac{2\pi n_{2}i}{\ln(2)}+\frac{\ln(\frac{3}{5})}{\ln(2)}}\text{ and }z=2^{\frac{2\pi n_{1}i}{\ln(2)}-\frac{2\pi n_{2}i}{\ln(2)}+\frac{\ln(\frac{3}{5})}{\ln(2)}}$$

$$\exists n_{2}\in \mathrm{Z}\text{ : }\left(\exists n_{1}\in \mathrm{Z}\text{ : }z=2^{\frac{2\pi n_{1}i}{\ln(2)}-\frac{2\pi n_{2}i}{\ln(2)}+\frac{\ln(\frac{3}{5})}{\ln(2)}}\right)$$

$$\exists n_{2}\in \mathrm{Z}\text{ : }\left(\exists n_{1}\in \mathrm{Z}\text{ : }z=2^{\frac{2\pi n_{1}i}{\ln(2)}-\frac{2\pi n_{2}i}{\ln(2)}+\frac{\ln(\frac{3}{5})}{\ln(2)}}\right)\text{ and }\exists n_{2}\in \mathrm{Z}\text{ : }\left(\exists n_{2}\in \mathrm{Z}\text{ : }\left(\exists n_{1}\in \mathrm{Z}\text{ : }\left(\exists n_{2}\in \mathrm{Z}\text{ : }\left(\exists n_{1}\in \mathrm{Z}\text{ : }a=2^{\frac{2\pi n_{1}i}{\ln(2)}-\frac{2\pi n_{2}i}{\ln(2)}+\frac{\ln(\frac{3}{5})}{\ln(2)}}\right)\right)\right)\right)$$

$$\exists n_{2}\in \mathrm{Z}\text{ : }\left(\exists n_{1}\in \mathrm{Z}\text{ : }z=2^{\frac{2\pi n_{1}i}{\ln(2)}-\frac{2\pi n_{2}i}{\ln(2)}+\frac{\ln(\frac{3}{5})}{\ln(2)}}\right)\text{ and }\exists n_{2}\in \mathrm{Z}\text{ : }\left(\exists n_{2}\in \mathrm{Z}\text{ : }\left(\exists n_{1}\in \mathrm{Z}\text{ : }\left(\exists n_{2}\in \mathrm{Z}\text{ : }\left(\exists n_{1}\in \mathrm{Z}\text{ : }a=2^{\frac{2\pi n_{1}i}{\ln(2)}-\frac{2\pi n_{2}i}{\ln(2)}+\frac{\ln(\frac{3}{5})}{\ln(2)}}\right)\right)\right)\right)\text{ and }\exists n_{2}\in \mathrm{Z}\text{ : }\left(\exists n_{2}\in \mathrm{Z}\text{ : }\left(\exists n_{2}\in \mathrm{Z}\text{ : }\left(\exists n_{1}\in \mathrm{Z}\text{ : }\left(\exists n_{2}\in \mathrm{Z}\text{ : }\left(\exists n_{2}\in \mathrm{Z}\text{ : }\left(\exists n_{1}\in \mathrm{Z}\text{ : }\left(\exists n_{2}\in \mathrm{Z}\text{ : }\left(\exists n_{1}\in \mathrm{Z}\text{ : }b=2^{\frac{2\pi n_{1}i}{\ln(2)}-\frac{2\pi n_{2}i}{\ln(2)}+\frac{\ln(\frac{3}{5})}{\ln(2)}}\right)\right)\right)\right)\right)\right)\right)\right)$$

$$n_{2}\in \mathrm{Z}$$

Leystu fyrir x, y, z, a, b, c, d

$$d=2^{\log_{2}\left(\frac{3}{5}\right)}\approx 0.6$$